For those interested in the history behind this proof, you may want to start here.
The proof presented is based on two books: Harold M. Edward's Fermat's Last Theorem: A Genetic Introduction and Paulo Ribenboim's Fermat's Last Theorem for Amateurs.
Theorem: Proof for FLT: n=5
x5 + y5 = z5 → xyz = 0 if x,y,z are integers.
(1) Assume there is a solution where xyz ≠ 0
(2) From previous results, we know that we can assume that gcd(x,y,z)= 1 (see here).
(3) We can also assume that x,y are odd and z is even since:
(a) We know that at least two of the values are odd since gcd(x,y,z)=1 tells that only 1 at most can be even.
(b) If two are odd, then the third is even since odd + odd = even and odd - odd = even.
(c) If z is even we are good so let's assume that x is even.
We know that there exists z', x' such that z' = -z and x' = -x so x' is even:
(-1)5(x')5 + y5 = (-1)5(z')5
First, we add (z')5 to both sides to give us:
(-1)5(x')5 + y5 + (z')5 = 0.
Then, we add (x')5 to both sides to give us:
y5 + (z')5 = (x')5
Even in this case, we have derived a form z5 = x5 + y5 where z is even.
(4) We can assume that 5 divides xyz from Sophie's proof.
(5) Now, it can be shown that if 5 divides z, then there is no integer solution (see here for proof) and likewise, it can be show if 5 doesn't divide z, then there is no integer solution (see here for proof).
(6) So we have a contradiction and can reject our initial assumption.