How does analyzing help me improve? This has been really frustrating for me since I don't get it. When I go through my games I spot errors and simply find better moves but then that's it. I just don't understand how this improves your game, all I did was find mistakes and correct them in a game I've already played. People keep telling me this is the best way to improve but I can't seem to get "it". If anything I suppose going through your games can show you where you might be lacking but wouldn't simply just continue learning and training in all areas of chess be a far more efficient way to improve instead of wasting time analyzing? I would appreciate examples of how analysis helped you become a better player.
First of all you have to distinguish which kind of error you made.

Opening error: 

You should definitely study opening theory to avoid opening traps or get bad positions right out of the opening. On the other side when an opponent plays an unusual/bad move, you need to know how to respond and punish this move to get a better position.

Middlegame error:

Have you overseen some tactical possibilities for you as attacker/defender?

-> Try to solve tactical puzzles to get familiar with certain positions. This will also help you to spot tactics more easily the next time they occur.

Have you had the wrong plan?

-> Study positional topics like pawn formations, weak/strong points, etc. in a position and you will find the right plan straightaway.

Endgame error:

When you have no clue how to play certain theoretical endgames you will easily take a wrong step and lose drawish positions or draw winning positions.

-> Take a good endgame book to study some basics.

Also learning from mistakes you make, means you should not make the same mistake again. Thats especially true for opening errors and endgame errors in easy theoretical endgames.