Analytic philosophy" can refer to: A general philosophical tradition characterized by an emphasis on clarity and argument (often achieved via modern formal logic and analysis of language) [emphasis mine]
What are some examples of how analyzing language can help solve philosophical problems?
EDIT: It strikes me that the analysis of language is analogous to a method of proof in math. For example, someone might ask me: "Why is it useful to analyze the group structure of arithmetic?" I could tell them Lagrange's theorem, and how Euler's theorem follows, and even if they didn't understand the first thing about groups, I believe I could convince them that the idea could be useful.
So what's an example of a problem that seemed difficult in philosophy, but the answer was illuminated by the analysis of language?