Comments from feedback form - "So it may be shrinking. Its ma..."
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Indeed, an increase in density would not lead to an increase in gravitational force felt at any given distance from the moon, as long as the distance was measured from the centre of the moon, and not the surface. Gravity is dependant on mass, not on density. However, *surface* gravity on the moon would increase ever so slightly as a result of the moon's slow contraction over time. How much gravity an object possesses is dependant only on its mass, but the strength of the force of gravity tugging on you is dependant on both the mass of the object you're near and your distance from it.
Because the moon is more dense now than in the past, while standing on the surface you'd be closer to the centre-of-mass of the moon, and therefore experience more gravity. (Note that if you dig into the moon (or any other object) you don't experience increased gravity because the material on top of you is now pulling you in the opposite direction.)
The same effect is what makes black holes interesting. If, for instance, you crushed all the mass in the sun into a black hole, the orbit of the earth would be unaffected. The mass of the sun (now a black hole) would be the same, so the gravity output would be identical. But stars are huge and diffuse, so you can't get very close to them before you start burrowing into their "surface", at which point only the portion of the star's mass below you would be tugging you in the direction of the core. The stuff on top of you would be pulling you upward. But with a very dense object like a white dwarf, neutron star, or black hole, their high density allows you to get very close to the centre-of-mass without anything pulling you in the opposite direction. Because you can get so much closer to the center, you experience a greater force than would be possible if the object were less dense.
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Referring to your quote "that if you dig into the moon (or any other object) you don't experience an increased gravity because the material on top of you is now pulling you in the opposite direction". Given that scenario, the mass at the center of the earth has all of the mass of the earth outside of it and should therefore be pulled "in the opposite direction". This implies that the earth should be hollow. Isn't this the kind of thinking that led to the "hollow earth" hypothesis. In fact, you can still become a member of the Hollow Earth Society. However, it is currently understood that the density of a massive object is highest at the center. Gravity is a function of both mass and density acting from the center of the mass "as a whole".
No, it doesn't imply that the Earth is hollow. What it implies is that if you dug a tunnel straight through the entire Earth (and used... I dunno... a forcefield to keep it open), at the very centre of the Earth you would be weightless, because all of the Earth's mass would be tugging on you equally in every direction.