Guide to Incrementals/Appeal to Players
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This is something that has been discussed and analyzed by many people, and to some extent, I feel like everything that can be said on the topic already has. However, a lot of these analyses are from the perspective of those with not as much experience and involvement within the genre as I'd argue would be necessary for a fully contextualized answer. I'm interested in ludology and part of that includes interpreting games as art, and to that end what constitutes a game, let alone a "good game". Incremental games are oft criticized, unfairly in my biased opinion, of not even constituting games, such as was posited by this polygon article.
Numbers Going Up
This is a very common response to why people enjoy incremental games, although it's not one I find compels me personally, and I suspect it might be a stand-in for progression) or Guide to Incrementals/What is Content?. But reportedly, some people do just like seeing big numbers. I must reiterate I suspect the actual cause is seeing big numbers in context though - if you start at 1e1000 of a currency and get to 1e1001, that isn't going to feel as satisfying as going from 1e10 to 1e100, and in any case, I don't think a button that just adds a zero to your number will feel quite satisfying - I believe its the sense of having made progress, and comparing where you are to where you started and feeling like you've earned your way here that is enjoyable.
Progression
I think a strong sense of progression is seen as very enjoyable to many players of all sorts of genres - engine builder board games, RPGs, rogue_lites_, etc. Incremental games tend to have an extremely exaggerated sense of progression, which makes them very appealing.
Meta-progression is when games have some sort of progression that persists when other progress gets lost - for example, upgrades that persist between runs of a roguelite game. These are common mechanics in incremental games - in fact, its not uncommon to have multiple of these reset mechanics nested on top of each other, each with their own meta-progression. These are satisfying to players, although they can be a bit controversial. These mechanics can often be seen as an optional crutch, and in roguelite games, players often challenge themselves to win without any meta progression. Essentially these challenges argue that meta-progression de-emphasizes player skill by replacing it with time served. Incremental games, through their exaggerated progression, eschew that possibility though - they make it impossible to beat without the meta progression systems, as the meta-progression becomes an entire chapter of the gameplay. I'd argue this does not detract from the game, however, and is actually a part of what makes incremental games, and roguelikes, enjoyable to many players: meta-progression augments the increases in skill the player is naturally gaining as they play. In effect, it's not replacing the skill increase, but exaggerating it to make it feel more real to the player.
Effortlessness
Incremental games are so easy, a lot of them even have you progress while you're not playing! Part of the appeal is being able to feel like you're making progress while doing something actually productive - multitasking, in a way. In this sense, the game is more of a fidget toy - not something to think hard about and play actively, but something to click a few buttons every so often while you're paying attention to a lecture or studying or working. Of course, not all incremental games lend themselves to being played this way - it's specifically "idle" games that work like this. These are games that take an incredibly long amount of time to see all the content, stretching it as thin as possible, but they aren't expecting you to be sitting at your device playing it the entire time. They expect you to leave and come back later to make a bit of progress and repeat the cycle.
If you look at the higher-level play of most games, you'll see them perform difficult feats with ease and speed. They'll achieve a "flow state" that takes all their knowledge and experience of the game and uses it to play the game as instinctively as possible. It's incredible to watch things like Slay the Spire speed runs or competitive DDR-likes. I'd argue the goal of a lot of games with a competitive scene is to get so good that the game becomes effortless. In that sense, a game that allows you to reach that point earlier isn't any less legitimate, but rather lowers the barrier to entry by allowing more people to get "really good" at the game. And to be clear, (most) incremental games aren't trivially easy - they, and to an extent, every game will have some level of learning and improvement over time.
Addiction
A lot of these reasons for why incremental games appeal may have reminded you of why gambling appeals to people, particularly those prone to addiction. Indeed, incremental games are quite often criticized for their similarity to a skinner box. Some have gone as far as to say incremental games as a genre are commenting that all games are skinner boxes). The argument goes that some games are not fun, but rather condition players into continuing to play without actually getting anything from the experience. When tied to real-world money this is seen as predatory, and to a lesser extent, even free games may be feeding the addictive sides of people and making them more prone to seek out gambling or micro-transaction heavy games.
While incremental games can be fun and even healthy in certain contexts, they can exacerbate video game addiction more than other genres. If you feel like playing incremental games is taking priority over other things in your life, or manipulating your sleep schedule, it may be prudent to seek help. See r/StopGaming for resources.
Since incremental games are often built on extrinsic motivations in the form of progression systems, it's hard to argue whether players continue to play because they are enjoying the gameplay, or if they are just conditioned to keep doing it because the game keeps rewarding them. Unfortunately, it can often feel like it's the latter, as there isn't typically anything compelling about the "gameplay" of clicking a button and waiting. There may be a significant overlap between those who enjoy incremental games and those who are most prone to addiction, and there are often posts on r/incremental_games about someone either struggling with or overcoming video game addiction.
Strategy
Incremental games could be considered a subset of strategy games), and inherit the appeals of strategy games. This includes the appeal of feeling like you've found a good solution to a puzzle, or that you're learning more about the game and are improving at making decisions within it.
Note that strategy games are not all the same difficulty, as well. Cookie Clicker is probably easier than Starcraft 2 (although late game may beg to differ). Plenty of incremental games can be used as evidence that "easier" strategies may have their separate appeal to harder strategy games - players like to feel smart and that they figured the game out and have optimized or mastered it, and the game being easier doesn't detract from that sense of accomplishment as much as it allows more and more users to be able to reach the point where they gain that sense.
Avoiding Staleness
Incremental games tend to have "paradigm shifts", where the gameplay changes in a meaningful way at various times throughout the progression of the game. These upset and change the gameplay loop, which helps keep them from stagnating. This constant "freshness" to the gameplay can keep players engaged for longer, compared to a game with a repetitive and static gameplay loop.
Good Game Design
Incremental games tend to show their game design "plainly", so it's more readily apparent if a game has good game design while playing, even if you're not looking for it. While different players have different preferences and might enjoy different types of games more than others, there are underlying good and bad game design principles that players will notice the effects of. To be clear, this isn't talking about stuff like big numbers being enjoyable, where I can comfortably agree to disagree with other players. They don't intrinsically make my experience better, but I'm aware of those for whom it does and I won't argue against their feelings. However, the game designer in me does feel like there are some extremely clear-cut examples of good and bad game design philosophies.
Let's start by giving an example of a mechanic I think can be easily and strongly argued is good game design. There are of course many examples, but a personal favorite of mine is how DOOM encourages aggressive gameplay by linking health drops to melee attacks. It has an intended experience it's trying to give the player - immersing themselves as DOOM guy, who would not hide behind cover when low on health - and this mechanic does a great job at encouraging and effectively teaching players to behave properly. This is in sharp contrast to shooters like Call of Duty, which have you regen health passively, encouraging players to hide behind cover and wait after getting hit. Note that I'm not arguing CoD is poorly designed, as the games have different intended experiences. I'm specifically praising DOOM for having a mechanic that does a good job at ensuring the player has that intended experience.
To contrast with an example I think is bad game design, let's talk about shields in souls-likes. This is a bit of a famous example, and I highly recommend this video essay which spends quite a good bit of time on this topic. Essentially, the argument boils down to players of earlier games in the souls games using shields too much - playing slowly, conservatively, and ultimately having less fun. Players wanted to feel safe, so they ended up playing in a way that ruined the experience for them. The developers solved this by removing shields, apart from an intentionally bad one effectively mocking the playstyle, and it did its job at getting players to play more aggressively, and often have more fun.
To bring the conversation back to incrementals, I'm incredibly opinionated on what makes a good incremental game, which I'll discuss in the game design section. Suffice it to say, incremental games rely more on good game design than other genres, due to not having much to distract from bad game design. This helps (although imperfectly - gamers are a bit too tolerant of bad game design!) well-designed games rise to the top within the genre.
Artistic Merit
The discussion of whether video games are art has resulted in a pretty universal "yes, they are", but with some games the argument may still crop up. The reason why Incremental games are sometimes questioned is due to their perceived lack of complexity. However, even setting aside the fact that if players are having fun then it's not time wasted, I think games can have artistic merit that supersedes the necessity of having (any / engaging / "deep") gameplay. Incremental games are no less legitimate of a game or the "art" label because of any lack perceived lack of depth. For what it's worth, most art can be consumed with more ease than any video game - any painting, movie, sculpture, etc.
A lot of incrementals have a narrative context that can similarly qualify them as art. Cookie Clicker is, as has been pointed out numerous times before, commenting on excess and increasing production beyond any reasonable limits - devolving into increasing production for its own sake. Indeed, a lot of incremental games are written to comment upon various concepts like capitalism or tropes in games, as discussed when defining Incrementals). However, I'd like to argue most incremental games are still art, even without any narrative context. "Art" as a concept is pretty nebulous already, but I personally like those who define it as an act of expression more than any physical result. The creator and the context within which they created the art, and any meaning they put into it, are all relevant and a part of the art itself. Most incremental games have artistic merit from things like why the creator made it, why they chose to make it an incremental game, and why they made any particular design decision. Hell, even if you play through an entire incremental game without a single thought or feeling, that very fact it elicited nothing can itself be artistic merit!
I'm not an art major, and I may be taking a somewhat extreme take on what is art and what has artistic merit, but I'd argue the overall point stands that games, and incremental games specifically, can have artistic merit, which appeals to many gamers.