It's not necessarily that you're dumb - I suspect you aren't. The problem is that you're trying to get a grip on these techniques when you haven't really mastered the absolute basics of algebra.

It's not surprising - you're trying to teach yourself, and it's always going to be that there are gaps. You need properly constructed tutorials and exercises.

Have a look at these:

http://www.google.co.uk/#q=algebra+tutorial+with+exercises

Pick one of the tutorials and work through it completely. Then you can do a google search for partial fraction tutorials and do the same.