Reading this book aimed at the layman gave me some understanding as to what exactly Wiles achieved by bridging different fields of mathematics to prove Fermats theorem. But it was so long ago, that all I remember is the general idea and no specifics. I am by no means a mathematician.

The general idea is that by showing mathematical equivalence between two fields that are seemingly completely distinct, something that is impossible to prove in one field but easy to solve in the other field effectively gets proven through this equivalence. The Langlands program's mission is to prove these kinds of equivalences between fields. Hence the use of the Rosetta stone in the article i originally linked.

EDIT:

ChatGPT's input:

The Langlands program is a far-reaching set of conjectures and theories that seeks to establish deep connections between number theory and geometry through the framework of representation theory. One of its main goals is to demonstrate relationships between Galois representations (which arise in number theory) and automorphic forms (which are related to harmonic analysis and representation theory).

Andrew Wiles's proof of Fermat's Last Theorem is a remarkable example of this interplay. Wiles showed that proving Fermat's Last Theorem could be reduced to demonstrating a particular case of the Taniyama-Shimura-Weil conjecture, which is part of the Langlands program. This conjecture posits that every elliptic curve is related to a modular form. By proving this relationship, Wiles was able to prove Fermat's Last Theorem indirectly.