Elegance is more of a gradient than a specific thing. Your game can be more or less elegant and different players will have a different tolerance for things. But ideally you want to do everything you can to remove barriers from the players’ enjoyment of your game. Here is a list of things to do to improve your game’s elegance.
1) Remove the not fun work from the players.
2) Everything should be clear, both visually and conceptually.
3) The more systems in your game are interconnected the more cohesive it will feel.
4) Incorporate the rules of the game on the components that use them.
5) Don’t let the game flow be interrupted.
6) Cut the boring parts.
In short, elegance is a player always knowing how to do what they want to do, and being excited to do it.
Final Tip
Remember that you (meaning all people working on the game) are in complete control of your game design. You can solve problems from multiple angles. If a rule is really hard to explain within the constraints of the design, you have a few options. You can do your best to explain it. You can change the constraints of the design. Or, you can change or remove the rule. The primary goal should be to make a complete game that works together as effortlessly as possible.
Good luck designing.