I have done Karate, Kickboxing and Boxing, but I have never learned about angles. I have only learned about striking techniques, blocking, parrying, combo attacks, countering with combo's, slipping + counters, bobbing and weaving + counters.
How does this superior angles thing work?
There's a lot of theory behind angles, but you don't really need to understand it all--especially if you're a fighter. Basically an angle is when you're lined up to hit the other guy, he isn't lined up to hit you (not as hard or as easily, he'll still always be able to hit you). It's about where both fighters' feet are. Angles are kind of like making your opponent stand in a shitty stance, either too square (inside angle) or too bladed (outside angle).
The most important thing about angles is you don't try to take big ones. Angles are about subtlety and minimalist movements. If you take a huge side step and pivot, your opponent is usually gonna just do the same and nothing changes. For an example of this, watch the footwork in Aldo vs Lamas. Lamas is trying so hard to circle around Aldo, Aldo just keeps making small adjustments and keeps him in front of him. Rumble vs Davis was the same thing. Davis literally tries to run to angles and it gets him nowhere. If you make it more subtle, you can be ready to attack again quicker so you can either catch him as he over-adjusts, or maybe even move farther than he notices so he lets you have the angle. Angles can be used when defending to eliminate the opponent's ability to simply swarm or chase you or when attacking to give you a better position to hit from.
Some examples: Jab/double jab as you take a half-step left and pivot, throw the right straight inside their lead shoulder. Probably the most common one.
Slip outside a jab, pivot as you throw your cross, attack from his left side. If you do this against a southpaw left straight, it's a very potent shot.
Your opponent tries to rush straight at you, pivot clockwise and set your feet. He's now square. This one is often attempted with a check hook.
Your opponent throws a right kick, you step back and let it miss, you now have an angle.
Angles exist naturally in a lot of situations too. Fighters will make footwork mistakes and give up angles so you don't have to take them. For example, a lot of fighters will miss a right hand then drag their right foot forward and give up an inside angle.
Again, there's a lot of reasoning behind why angles are so effective and important but a lot of it is only necessary if you plan on coaching or really need to be able to think something through to learn it. If you just get that you want your feet lined up in a good stance while the opponent's aren't, you're good.