This is because as you get inside the earth, you are not only pulled towards the center but also attracted by matter "behind you", i.e. there is a force counteracting the motion towards the center. The result a decrease in the total acceleration as you move towards the center, and as you reach it, it should be zero (provided you manage to stay there without moving). To visualize this, here is a diagram showing the gravitational potential energy in the presence of a uniform sphere, which approximately represents the earth:
In the picture, $a$ represents the radius of the surface. As you can see, inside the sphere, the energy (and the force) decreases linearly as you approach the center. The inverse square rule only describes the behaviour of the field outside the sphere. Note that the linear behaviour is just valid in the approximation of uniform density, which is not realistic. This answer should however give an idea of what happens to the gravitational field inside a spherical body.