How to interpret the gradient of implicitly defined function like f(x, y, z) = 0?

Question

I know the definition of the gradient for 2 input function like f(x, y) = z. The gradient will give out a 2 dimensional vector that represents the steepest ascent. But if I subtract the z from both side, I'll get the function f(x, y, z) = 0. The gradient of that function is now a 3 dimensional vector, what's the meaning behind the new gradient?