Since:

(x+5)^5 mod 25 = x^5 (expand it out to prove)

and 1^5=1, 2^5=7, 3^5=18 and 4^5 = 24 (mod 25)

it follows that x^5, y^5 and z^5 = 1,7,18 or 24. for any x,y,z not divisible by 5.

Now we cannot add two numbers from 1,7,18 and 24 to get 1,7,18 or 24 so the equality x^5+y^5=z^5 is false.