That equation applies for point sources, which the Earth technically is not. We can, however, treat the Earth as a point source as long as its internal structure is irrelevant (i.e. as long as we are outside of it). Once we enter the surface of the Earth, we can no longer simplify it by pretending it's a point and we have to perform a full analysis of the system.
It's important to note, as joshphysics pointed out, that because the density of material in the Earth isn't a constant, moving deeper underground will actually put us closer to a region that is denser (and therefore pulls on us more strongly) while putting a region that is less dense farther away, which would increase the force of gravity. For a first approximation, however, we can assume that the density of the Earth is constant.
So, given this assumption, we can show that only the amount of mass that is still underneath us actually exerts a net pull on us in a variety of ways.
For instance, it's a known result that at any point inside a spherical shell of mass, there is no net gravitational force due to symmetry. We therefore know that once we are inside the Earth, any mass that isn't as deep as we are has no net effect on us. One can also use Gauss's Law to demonstrate the same thing.
In short, the only mass that exerts a net force on us is the mass that is below us, and the deeper we travel underground, the less mass is beneath us. Therefore, there is less gravitational pull as we travel deeper beneath the surface.