Yes, philosophers tend to be interested in the issue of conceptual analysis. Different philosophers will have a different understanding of what conceptual analysis is but one story goes something like the following. First, we start out with a rough, intuitive sense of the concepts that we use and this gives us a series of criteria for each concept (perhaps with free will one criteria would be that it relates to moral responsibility in some way and another would be it relates to the ability to do otherwise in some way). Then we try to find a more precise account of the concept that does the best (though not necessarily perfect) job of satisfying these criteria.

I personally find the level of focus on conceptual analysis in philosophy frustrating so I'm not sure that I can do justice to a defence of it. I know many very intelligent people who think it is indispensible to our reasoning though so it may well be deserving of further reflection. If you're interested in such reflection I suggest that you read Frank Jacksons, "From Metaphysics to Ethics: A Defence of Conceptual Analysis". It's a short book and gives a good sense of how contemporary philosophers think about conceptual analysis (in terms of what conceptual analysis is, btw, Jackson says the following: "The short answer is that conceptual analysis is the very business of addressing when and whether a story told in one vocabulary is made true by one told in some allegedly more fundamental vocabulary.")

Off the top of my head, why might someone think conceptual analysis is important? First, conceptual analysis is all about getting clear on our terms. If you're discussing free will, it seems like a really bad idea to just debate without making clear what you mean by free will. So it seems useful to get clear on our terms.

Myself, I'm tempted to say we should get clear on our terms by stipulation (though note that even this involves a small amount of conceptual analysis or I would be just as likely to stipulate that "free will" means "eating my hat" as I would be to stipulate that it means "my decisions flow from my deliberative process": and many philosophers only use conceptual analysis in this easily-defendable manner). So I would stipulate what various meanings of free will are, say which we do and do not have and leave it to each individual to figure out how that makes them feel (which type of free will they care about).

However, a lot of people don't find this very helpful. They care about FREE WILL or MORALITY or BEING THE SAME PERSON TOMORROW (a.k.a. personal identity). And they need to know what the best realiser of the criteria that they have for this concept is to know what they care about. So if you tell a lot of people that they have free-will-1 and not free-will-2, they don't care about this: they care whether they have FREE WILL and so they need to find out which of free-will-1 and free-will-2 is a better realiser of their concept before they can work out whether they have the sort of free will that they care about. Insofar as I don't think telling people what they should desire (no, you should desire free-will-1, regardless of the nature of your concept of FREE WILL), I don't really have any objection to the claim that such a person needs to carry out a more robust project of conceptual analysis (though I feel no need to join them on the road they're travelling).

All that said, standard epistemology (as opposed to formal epistemology) is one of the worst areas of philosophy to study if you're uninterested in these conceptual debates as such debates are pretty much the entire of the field (where in many other areas of philosophy they play a smaller, though often still substantial, role).