ELI5 version of this madness:
"If we have a Galois group, then we can look at its representations. Galois Representations are really important, but they are never done in isolation. Wiles did a whole bunch of work to show that for ever Galois Representation of a certain type, you get a modular form that "matches" it. This means that you can use what you know about modular forms to conclude something about these kinds of Galois representations (eg, that there is no Galois representation of our type with a certain signature needed by a counterexample to FLT).