Almost always, when I talk with groups of people about game design, I quote Antoine de Saint-Exup'ery: "A designer knows he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away."
Another way of saying this is about Japanese art-gardening "Your garden is not complete until there is nothing else that you can remove."
I make games that are models of something, or are “pure” abstract games, that is, games that are very simple in conception. But there is an opposite philosophy of making a puzzle-like game more complex so that the puzzle is harder to solve. Simplification is quite a different matter in that situation, and something I can't address specifically both because it's the opposite of my philosophy and because I dislike that kind of so-called "game."
I try to be simple throughout, but it's common for designers to start with something simple and keep adding things to it, until they realize that they weighed the game down too much and need to go back toward the simple. When I see a problem in my design I try to find a simple solution, possibly a simplification, rather than add something to the game, but many designers will usually add something to fix a problem. And then those additions can become too much weight.
As I answered questions after a session at a convention, someone told me about an RPG he'd designed and tested, that all the testers said was too complex. "How do I simplify it?" he said.
An assumption here is that the testers, by and large, aren't able to say exactly what must be simplified, they just know that currently there's "too much."
First, I said, try to write down the "essence" of the game in a few sentences. This can take some doing, believe me. Ideally you've done it already, but if you had, perhaps you wouldn't be having the too-much-complexity problem to begin with.
There are different ways to characterize the essence of a game, sometimes structurally, sometimes according to what the player does, sometimes in another way or a combination. But be sure it's a unique essence, not just a list of mechanics, because the list the mechanics doesn't say anything about what's important or what the impression on the player is supposed to be.
Example (Britannia): "On an anvil of blood and terror they forged the destiny of an island!" In this epic wargame four players each control several nations playing at different times with different objectives throughout the Dark Ages history of Great Britain. Romans, Britons and Gaels, Anglo-Saxons, Vikings, Normans - they all play a part in the history of Britain. Combat is resolved with dice. This is a strategic game of achieving objectives, not of conquest, though many invaders conquer large parts of Britain at different times. 4 to 5 hours for experienced players. "Invade Britain. Rewrite history. Rule." (The quoted phrases are tags devised by the publishers of the first and second editions of the game.)
Then think about the various aspects of the game in relation to that essence. If something doesn't contribute to the essence, can it be removed? Surely, at the least, it can be simplified, abstracted, or combined with something else. Every game (tabletop or video), at bottom, is fairly simple, and your job is to retain its simple heart and remove what doesn't contribute enough to that heart.
Second, make a list of the major features or elements of the game, perhaps 10 to 20 of them. Consider again how they contribute to the essence, and how you can remove or simplify or combine as appropriate.
After you have (in your mind, at this point) removed or simplified what you confidently can, give the list of the (now remaining) major features to some of your playtesters and ask them to decide which could be removed entirely, and which should be simplified. (This may not help much if testers disagree about whether the game is too complex.) Don't ask people to rank each feature in comparison to the others, as that can be very difficult. It's much easier for people to divide a group into four parts, in this case from most important to the game down to least important. You might even want to write each feature on a separate 3 by 5 index card to make it easy for the playtester to sort them.
Whether you ask playtesters individually or in groups depends on what you think they'll be most comfortable with.
Then consider how you can get rid of the items in the bottom quarter, or even the bottom half if the game is much too complex.
Then playtest the result, of course.
I've listed these in an order beginning with what you can do on your own, to what you can do in conjunction with your playtesters.
As for some details:
Perhaps you can simplify a game by combining two things together into one. You don't actually eliminate either, you just streamline.
An obvious way to simplify is to find decisions players must make that have no significant effect on the outcome of the game. In other words, if it doesn’t matter what the player chooses, if it's trivial, why make them choose at all?
Similarly, there can be choices for particular decisions that no one ever chooses. Get rid of them, if that's possible.
You can resort to abstraction, that is “remove a more accurate and detailed version of that aspect or function and replace it with a less accurate and detailed version or no version at all” (Adams and Rollings, Fundamentals of Game Design).
Automation is often a means of simplification in video games. While we cannot exactly automate anything in a purely tabletop game, we can take something onerous for the players, and turn it into something that happens in the background or automatically, from their point of view. (Yes, some tabletop games include smartphone apps to automate.)
Some games, including virtually all RPGs, are models of some situation, even if it’s a fictional situation as is often the case in RPGs. When you’re modeling something it’s a little easier to simplify, I think, because when you take something out of the game you can try to judge how that affects the correspondence of the model to the situation. When you’re simplifying an abstract game, and that includes a game with a tacked on theme such as many Eurostyle games, then you don’t have that guide.
In either case, harmony has to be one of your guides. I discussed this at length on Gamasutra (https://www.gamasutra.com/blogs/LewisPulsipher/20170424/296624/Harmony_and_the_Kludge_in_Game_Design.php) and I’ll try to summarize here. Harmony: “everything in the game feels as though it belongs there and contributes to the purpose and feeling of the game as a whole.” This is like music harmony, it’s easy to recognize but not so easy to create. It's important because games are not just collections of mechanics, not just data, not just metrics. Games make intellectual and emotional impressions on players, and lack of harmony is noticeable, sometimes clearly, sometimes in subtle ways.
So the question is, what can be removed because it is inharmonious with the rest of the game? Or when you’re going to remove something, what will is do to the harmony of the game? And that’s going to apply whether the game is abstract or a model.
Sometimes a game's interface will get in the way, much more often true with video games than with tabletop games. Is there a way to make the interface simpler, to provide information in more accessible ways or to make it easier to manipulate the game?
Let me give you a drastic example of simplifying a well-known game: Monopoly. Unless there are few players in the game, it's standard practice to buy whatever property you land on as you go around the first few times. This becomes a trivial decision, especially because most people incorrectly play the game without auctions; so it should be removed from the game. Why not shuffle the property cards and deal three or four to each player at start of the game, and let them decide whether to buy those properties or to put them into auction? (You might want to give players several hundred dollars extra because they're not Passing Go in this situation.) Once that's sorted out then you get on with the game. The game is not (should not be) about going round and round randomly, it's about trading properties and making monopolies. At least, that's what the game is for adults, for children the going round and round might be an enjoyable part of the game.
This simplification should make it a significantly better game, yet the effect of most simplifications is not as obvious. It's the accumulation of simplification that helps polish the game and get from the 80% point (well, it works) closer to the 100% point (it's a good game).
Always beware that you don't take something out that's essential to the game. While you can simplify a game into oblivion (a bagatelle), it's much more likely you will complicate a game into oblivion (a train-wreck). And always keep Harmony in mind.