Proving this is a classic problem in undergraduate physics.

If the mass is spherically symmetric, this will not be the case; all of the Newtonian forces from the masses further away from the center than you are will cancel out. This is called the "shell theorem", and it turns out to hold even in General Relativity.

For each point of any given distance, you can find a disk of points at the plane of the same distance whose gravitational pull will be equivalent to a single point at their center.

You do this for all distances x, and you will find an equivalent rod going from the attracted objected to the center of the sphere, of a non-uniform but symmetrical density.

We find that the density is linearly correlated to the area of the corresponding section, and we then take the closer half of the rod, multiply the density by 1/r^2, and find that it is constant!

For the far half of the rod, we cannot do this, so instead we divide it into two halves of equal pull, then divide those to halves and so on, and find that it this reduces to the equivalent of a point.

Now that we have two points in-line with the object, we find the point with the same total mass that exerts and equivalent force, and lo and behold it is at the center.

I'm sure there is a much simpler way to intuit it, though :)