A truth is analytic if its proof depends only on general logical laws and definitions.
So, for example: a bachelor is an unmarried man is an analytic truth. I am writing this sentence on a Macbook Pro is a synthetic truth.
Analytic philosophy really got started in the early 20th century with the development of quantification theory (formalized into something called predicate calculus). This new logical system was far more powerful than simple propositional logic. It is, for example, possible to ground arithmetic in predicate calculus, meaning that the truths of mathematics are analytic, rather than synthetic (as many earlier philosophers had held).
The only part of any of this that you need to know is that while obviously philosophers working outside of the analytic tradition (often referred to as "Continental philosophers," as the analytic tradition is largely an Anglo-American one) don't reject logic, they do reject the claim that formal logic is The. Tool. For. Examining. Philosophical. Claims.
Continental philosophers argue that the analytic sorts miss a lot of things, as their conception of what counts as philosophy is overly narrow. Analytic philosophers, on the other hand, argue that their colleagues in the Continental tradition accept propositions that don't hold up to formal logical analysis.
These are the traditional metaphors of analytic philosophy—the style that can be represented in the form of A is B.
Where analytic philosophy is an embrace of analytic truths and formal logic as the sine qua non of philosophical investigation, conceptual metaphor theory is a full-throated rejection of that idea.
I’m sure she meant it as a compliment. I certainly took it as a compliment, too. I mean, I studied analytic philosophy. Linear arguments are kind of our stock in trade.
A rigorous answer to that question would require more words than any sane person is willing to read, but the nutshell version is that analytic philosophers break claims down into their smallest constituent parts (be it terms, concepts, or propositions) and then analyze each part through the lens of formal logic.