From the custom puzzle menu it seems it can be all kinds of things.
What is the logic in solving puzzles?

I heard that, but I'm trying to learn. It feels like that point in learning a language when you're not quite fluent but nearly there. I just don't know how to push on through to the next level. I don't get the logic, but I want to.

I started doing chess puzzles because I investigated and found that again and again these were recommended for getting better at actually playing chess. The idea is that it improves one's ability to analyze the board.
I am getting extremely frustrated trying to solve puzzles. My puzzle rating fluctuates wildly between high 1800's to low 2200's but is usually somewhere above 2000's. When the puzzles are applications of mating patterns or even known capturing patterns, I can generally get them. Then there are other puzzles that seem to have no rhyme or reason to choose one move over another. The "solutions" seem almost completely random. Sometimes, they don't even obviously make the position actually better. At first I thought, "Oh, I was doing a 2 step checkmate but there's a 1 step mate I missed." Sometimes, that is what's going on. Sometimes, it's just not.
Supposedly these puzzles are helping me analyze the board better to play but I just feel like I'm beating my head against the wall. What is the way of looking at things beyond the simple checks, threats, and captures level?
Almost every puzzle should have some relatively obvious reason why it's better. There may be a very small percentage that are just bad puzzles and have no clear/satisfying solution, but evaluate as best, and those can be reported to be removed. However, those are relatively rare.
When looking at analysis, for those you don't see a clear reason, that should give you a better idea on what is going on.
There are different ways to do puzzles. You try to get them quickly and practice your pattern recognition. You can take your time and only answer when you have calculated the whole line and all the refutations. There are all types of puzzles in the database, so you'll often need to calculate to see what the answer actually is.

The point of a chess puzzle is that there is only one solution. The solution may vary from being a tactic to win a pawn or a piece, or an eventual checkmate. The problem is that you as the puzzle solver may not immediately know which of the resulting themes the puzzle is trying to train. In fact, puzzles are all about finding the absolute BEST move REGARDLESS of the position. This trains your brain to similarly aim to play the best moves in your chess games, hopefully at least in the tactics that the puzzles discuss. Although in a real game you might be able to checkmate in two moves even though you didn't see checkmate in one, in a puzzle this would be wrong because as I explained puzzles are designed to always have ONE and ONLY ONE definitively BEST solution; you always have to play the absolute BEST move in the puzzle to get it right, and if you don't think it's right, just check with the computer and it will tell you that you're wrong!

What is the way of looking at things beyond the simple checks, threats, and captures level?
I often analyze backwards... instead of looking at the position in front of me, I picture in my mind the final position that I want (a mating pattern, a fork, or whatever) and try to figure out how to get there from here.
Mating patterns should be studied by family, anyway... Greco mates, Lolli mates, Morphy mates, and so on.
I'll offer a few Lolli mates as an example of a family of checkmate patterns. Lolli mates are characterised by an advanced Pawn on f6 (or f3 for Black) which supports the mating attack.

Soccer is not about penalty kicks. Chess is not about puzzles. In a real game nobody tells you there is a tactic, or for which side.

...My puzzle rating fluctuates wildly between high 1800's to low 2200's but is usually somewhere above 2000's. When the puzzles are applications of mating patterns or even known capturing patterns, I can generally get them. Then there are other puzzles that seem to have no rhyme or reason to choose one move over another...
Supposedly these puzzles are helping me analyze the board better to play but I just feel like I'm beating my head against the wall. What is the way of looking at things beyond the simple checks, threats, and captures level?
Hi @whac3 there are a few things to address here. The first one is that the purpose of puzzles is not to analyze the board better. The purpose of puzzles is to expose yourself to more patterns (usually tactical) and this will help you cement these ideas into your subconscious "pattern recognition" with enough practice. This is a fancy way of saying: you are more likely to find these patterns in your own games if you've seen them somewhere before (like puzzles).
The next thing I'd like to address is that good puzzles only have one solution and this solution is favored over the other options by something significant. If you don't understand why the solution move is best, then you are missing out on learning opportunities. Next time you don't understand why the solution is best, put the position into the analysis board and try to figure out why your move is incorrect.
Lastly, I just want to say that you have a HUGE puzzle rating for your game ratings. I can tell you've done a lot of puzzles (and that's awesome!), but it's usefulness has probably exhausted itself for now. Chess is WAY more than just puzzles and you should probably learn some of those other elements of the game. When you bring up your game ratings some more, then you may revisit puzzles, but I think your puzzles rating is fine for a while.
In fact, my most recent YouTube video (excluding the YT short I recently made) lists solving only puzzles as one piece of bad chess advice. I recommend the entire video, but the 3rd point in the video deals with puzzles. See if you can find the best move in that position. If you have mostly worked on tactics puzzles, then you are unlikely to find the best move after ...Rfe8.

In your case, Yes, you are beating your head against the wall. Let me solve this issue, You should understand and remember the patterns instead of solving thousands puzzles. There are many things which i can not cover here, if you need detailed answer with solution, you may post your question at: thechessforum.com
I started doing chess puzzles because I investigated and found that again and again these were recommended for getting better at actually playing chess. The idea is that it improves one's ability to analyze the board.
I am getting extremely frustrated trying to solve puzzles. My puzzle rating fluctuates wildly between high 1800's to low 2200's but is usually somewhere above 2000's. When the puzzles are applications of mating patterns or even known capturing patterns, I can generally get them. Then there are other puzzles that seem to have no rhyme or reason to choose one move over another. The "solutions" seem almost completely random. Sometimes, they don't even obviously make the position actually better. At first I thought, "Oh, I was doing a 2 step checkmate but there's a 1 step mate I missed." Sometimes, that is what's going on. Sometimes, it's just not.
Supposedly these puzzles are helping me analyze the board better to play but I just feel like I'm beating my head against the wall. What is the way of looking at things beyond the simple checks, threats, and captures level?