To understand the proof, you need to have a good background in arithmetic and algebraic geometry (including but certainly not limited to the theory of elliptic curves), commutative algebra, algebraic number theory, and modular forms.
If you have an understanding of these things at a graduate level (say a second
or third year graduate student who is beginning research on these sorts of topics), then you will be able to understand the strategy of the proof and some
aspects in more detail. If you are at the level of having successfully written a thesis in these sorts of areas, you will be able to have a substantial understanding of the proof.
Having a complete understanding is perhaps even more difficult (as Kevin Buzzard points out in a remark in one of the MO threads linked to above) because the proof uses base change results in the theory of automorphic forms (due primarily to Langlands) which are quite
technical to prove, and rely on techniques quite different to most of the techniques that Wiles himself uses.
The standard introduction to the argument is the graduate text "Modular forms and Fermat's Last Theorem" (edited by Cornell, Silverman, and Stevens). Another is provided by the long article titled "Fermat's Last Theorem" of Darmon, Diamond, and Taylor. Of course there are many introductions at a more basic level (see the MO threads linked above), but the two references I've given actually contain many details of the proof.
The upshot is that it's probably not realistic to expect to understand much more than the very big picture strategy of the proof unless you are a graduate student focussing on this part of number theory.
Best Answer
Unless new historical documents are discovered, we can never know for certain what Fermat had in mind when he made his famous FLT remark. Most number-theorists probably share the same opinion as Weil (quoted below), that he made an elementary mistake, e.g. thinking that results for smaller exponents would generalize. Nowadays it is known that Fermat's Last Theorem cannot be proved by certain types of descent proofs similar to the classical simple proofs known for small exponents (search for "Tate Shafarevich obstruction").
Below is Andre Weil's opinion on this matter, from his historical treatise Number Theory, p.104.
Remark $ $ It is a common inaccurate hunch that shortly-stated theorems should have short proofs. However, this is easily disproved. For any formal system that has nontrivial power (e.g. Peano arithmetic) there is no recursive algorithm that decides theoremhood. Suppose that there existed a recursive bound $\rm\ L(n)\ $ on the length of proofs of a statement of length $\rm\:n.\:$ Then we could test for theoremhood simply be enumerating and testing all possible proofs of length $\rm\le L(n).\,$ Therefore there can be no such recursive bound on the length of proofs. Therefore there exist short-stated theorems with arbitrarily long proofs -- proofs so long that they are probably not amenable to human comprehension (this was observed by Goedel in his 1936 paper on speedup theorems).