Damn, alot of this is going over my head. I really don't understand the benefits of being at certain angles and what actually makes it an advantage to be at certain angles. I don't understand the ways to exploit your opponent from certain angles. It may be time for me to join a new gym.
Basically, you want your position to accomplish two things.
1) You can hit your opponent, but he can't hit you. Usually this is expanded to "you can hit him
cleanly and he can't hit you cleanly." You're rarely going to find an angle against a competent opponent from which he can't hit you at all, but the idea is to minimize risk to yourself and maximize risk to him. Which brings me to...
2) Your shots will do maximum damage. Often this means catching your opponent in such a way that his balance is compromised. For example, if you stand right in front of someone in a good stance and hit him with a right hand, the force of that punch will travel down his back, through his back leg and into the floor. This is because he was well positioned to absorb damage. In other words, his strong stance allows him to dissipate the force of your punch.
Now, if you move to his left, with him still facing the same direction as before, and hit him with the same right hand, he is much more likely to be hurt by it. Without his back foot directly opposite the force of your punch, his balance, and therefore his durability, is compromised, and a lot more of your punch will be absorbed by his head alone, rather than dissipating.
There are a ton of factors, but those are the two basic things you want to achieve when you move your feet. You're trying to hit him without letting him hit you, and you're trying to hurt him when you do.