Why does gravity have a stronger effect on objects with more mass? Doesn't moving something heavier require MORE force? Is it just one of those things where we just say "well that's how it works and here's some math to describe it"?
 A: In Newtonian physics one can consider two kinds of mass: gravitational mass (responsible for creating gravity) and inertial mass (responsible for resisting movement). We could certainly imagine that these could be different. But it appears that in our world they are the same: that the mass $m$ in $F=ma$ is the same as the mass $m$ in $F=G\frac{Mm}{r^2}$. Because they are the same,  they cancel, so we get $a = G\frac{M}{r^2}$, and thus the acceleration due to gravity does not depend on the mass of the object being accelerated. That is, a bigger mass needs more force to be moved, but also creates more force due to gravity by the exact same factor. You're right to wonder about this: it is a remarkable coincidence, and other forces do not work this way.
Einstein used this coincidence (the "equivalence principle") as one of the inspirations for his theory of general relativity. In relativity the equivalence of gravitational and inertial mass is not a coincidence at all. The "force" of gravity is a consequence of the shape of spacetime, and hence appears as a kind of "pseudo force" or "inertial force" similar to centrifigual force. Objects always try to follow geodesics (the general version of "straight lines") in curved spacetime, and for example near the Earth these geodesics are the paths followed by falling objects. While you're falling, you feel no force: only when you hit the ground do you feel it, and that force is due to the electromagnetic forces between the atoms in your body and the atoms in the Earth, which push you away from the curved space geodesic that your atoms are trying to follow. Now only the inertial mass (resistance to the force exerted by the Earth's atoms) needs to be considered, and hence relativity explains why the force we feel due to gravity is proportional to our inertial mass.
A: Gravity is an extensive property; at any given distance-to-center, the Earth's gravity is roughly 81 times that of the moon, because the Earth is that much greater (in the gravity-field-production property of 'gravitational mass').
By Newton's third
law of motion, the fact of Earth exerting a force on a brick ensures that
the brick also exerts an equal but opposite force on the Earth.
Thus, the gravity force on the brick must, logically, be proportional
both to the mass of Earth, and the mass of the brick.   That's the
only kind of proportionality that such a force could exert that fulfills
the Newtonian law.
Newton also identified mass as a ratio of force and acceleration, in his
second law.   That, is a discovery; it DOES apparently work that way,
but since acceleration is  not exactly the same as gravity, it is
an oddity.   The 'inertial mass' of the second law seems, to
a very close approximation, to be the same as the 'gravitational mass'.
