I have to disagree with Paul: Functional analysis is a beautiful, elegant topic with an enormous range of applications (but de gustibus... :)).
But in any case, you don't need to know a lot of (pure) functional analysis to understand finite element methods: Normed vector spaces, (strong) convergence, completeness, inner products and their induced norms, orthogonality, dual spaces and adjoint operators. Except for the last two, these topics should be already familiar if you already know some multivariate analysis and linear algebra (modulo some technicalities). They are all treated in (the first four chapters of) Kreyszig's Introductory Functional Analysis which was already recommended (and which I found to be a very pleasant read).
But much more than the behavior of abstract vector spaces, you need to understand the behavior of Lebesgue and Sobolev spaces, as well as the theory of (weak solutions) of partial differential equations. (From your previous question, I would say that this is precisely what you have problems with.) These topics (which some would classify as (applied) functional analysis) are usually summarized, but not treated in detail, in what I (as a mathematician) would call the fundamental and standard finite element books:
These are listed roughly in order of mathematical abstractness, and should be enough to be on the same page as the scientists you're talking to. If you want more detail, you need to look at books on the theory of partial differential equations; the standard references would be (again in order of abstractness)
In fact, Chapter 5 and 6, as well as Appendix D and E, of Evans' book should cover most of the theoretical background you need for the (mathematical) theory of finite element methods.