To "get" something means to understand it. For example, when you didn't understand something, but then your friend explains it to you in a way that you do understand, you can say:

Oh, OK. I get it now!

The phrase "I didn't quite get ___" means that you almost understood something, but you didn't actually understand it. The words "not quite" mean "almost, but not really". Another example of "not quite" is:

I'm not quite sure what I want to do after college.