If you want the negative solution to show up you have to write the equation with Surd:
Solve[Surd[6 x - 5, 3]^4 == 625, x, Reals]
or with CubeRoot
Solve[CubeRoot[6 x - 5]^4 == 625, x, Reals]
This is because Mathematica powers are otherwise meant in the complex sense, as the principal root: a^b=Exp[b*Log[a]]. When the base a is negative, the Log is nonreal. Compare
Plot[x^(1/3), {x, -1, 1}]
Plot[CubeRoot[x], {x, -1, 1}]
I always warn my students that the familiar algebraic rules break down when dealing with powers with a negative base and noninteger exponent.