What's going on with this 5-line proof of Fermat's Last Theorem?

The point is you could equally replace "FLT is true" with "FLT is false" or "My head is made of cheese". By introducing $R$, the author makes the system inconsistent and once a system is inconsistent, every statement in it is both true and false. This is called the principle of explosion.

Let $S$ be a statement which is both true and false. Since $S$ is true, $S \lor T$ is true for any statement $T.$ Since $S$ is false and $S \lor T$ is true, $T$ must be true. Thus $T$ is a true statement, i.e. every statement is true.

This is the idea that the "proof" is trying to get across. The statment that $R \in R$ is both true and false, and so using the above arguement, you can prove that anything is true, in particular FLT. Similarly, you can prove that any statement is false.