you know that y = g(x). it follows that g^-1(y) = x.

now suppose that a is in Stab(x).

then gag^-1(y) = g(a(g^-1(y))) = g(a(x)) = g(x) = y.

thus gag^-1 is in Stab(y).

similarly, if b is in Stab(y), then g^-1bg is in Stab(x), so

b = g(g^-1bg)g^-1 is in g(Stab(x))g^-1.

from this we can conclude that g(Stab(x))g^-1 = Stab(y).

(we only need the first part if G is finite, since then

g(Stab(x))g^-1 contained in Stab(y) shows they are equal).