I think the error here is that Analytic Philosophy can be succinctly summed up as though it was this tangible specific concept. There was a reaction against Hegel's grand, sweeping yet obscure proclamations about reality.
Such abstraction made philosophers realize just how much we take for granted regarding language and words. This led them to examine words and their relationships to meaning and truth much more closely.
However, it would be wrong to think that Wittgenstein's specificity and focus on the mechanics of language meant that this is 'the aim of analytic philosophy'. What he found was that language isn't logical and concrete, he found the opposite-- that language is absolutely not some objective thing to figure out. Words mean different things to different people at different times used in different ways. Essentially, even though language is complex, regarding ideas and concepts, we essentially use the same combination of a handful of rudimentary tools to accomplish almost everything regarding language and words.
You're a math guy huh? Which did you like better, geometry or algebra?
Mathematics are kind of a good analogy of different 'levels of reality' in philosophy...they each have their own kind of internal coherent logic, which is what math essentially is too, wouldn't you say?-- different levels and forms of coherent systems of logic?
As a mathematician I can see why philosophy a la Wittgenstein would be particularly appealing to you. . . math guys can sometimes
Just don't forget; Wittgenstein's nitpicking wasn't the point of what he was doing, it was merely the starting point of trying to always ensure as much clarity as possible. We don't do math for maths sake, we use math to create things and understand things, we don't own cars to work on our cars
Just as we find that on the very basic, smallest, most fundamental quantum levels of reality, everything(including the math) gets really weird and abstract, there are deep truths of life that don't neatly fit into a dense newtonian logic..
I hope I'm not being too vague here.