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If there was no gravity, you simply couldn’t get them to move without a great deal of effort. But for some reason, in the absence of any other force, just having them within a couple of hundred million miles away is enough for them to move together. This is kind of mysterious isn’t it?
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There are two “deep questions” here that are kind of woven together: action-at-a-distance, and the equivalence of gravitational & inertial mass. The idea of “action at a distance” was really first popularized by Newton when he came up with his original theory of gravity. To explain how two objects that were not obviously in contact could still exert a force on each other, he came up with the idea of a “gravitational field”: at every point in space, the field tells you how a mass will act under the influence of gravity. For example, if I’m near the surface of the earth, the gravitational field of the Earth is such that a mass will experience 9.801 Newtons of force for every kilogram of its mass, and that this force will be directed towards the center of the Earth.
You might protest that this doesn’t really explain anything; as I’ve described it, it just sounds like a convenient bookkeeping tool, but nothing more. As it turns out, though, the gravitational field turns out to have a life of its own; once you’ve figured out how to extend Newton’s theory of gravity to a more complete description of the Universe (Einstein’s General Relativity), you discover that the gravitational field has ripples and waves that can propagate through space on their own, without any need for mass nearby. So it certainly seems like the gravitational field should be considered an entity in its own right. (This is an analog of the link between electromagnetic fields and light, if you’re familiar with that.)
The other question is why the force of gravity seems to be so big; how is it that the gravitational field can act on a stupendously large mass to get it moving? This is where the equivalence between gravitational and inertial mass comes in. As John Mace noted, the gravitational force acting on an object is proportional to its mass; in other words, the more mass your object has, the stronger gravity acts on it. What’s more, Newton’s Second Law tells us that if a given force acts on an object, its acceleration (i.e. the amount it will move in response to the force) will be decrease as the mass increases. If you combine these two tendencies (with some not-too-complicated mathematics that I’ll spare you), you find that these two tendencies cancel out, and the acceleration that an object undergoes due to gravity is exactly the same, regardless of its mass.
This cancellation is due to the two notions of the mass — the “gravitational mass” which determines the force due to gravity, and the “inertial mass” which determines the acceleration due to a given force — being identical for all objects, at least to the accuracy we’ve been able to achieve with our experiments. This equivalence seemed somewhat fortuitous and mysterious until Einstein came along, with his aforementioned theory of General Relativity.
Einstein’s rather radical notion was to say that gravity isn’t really a force at all; rather, it’s a so-called “fictitious force”. A more easily comprehensible example of a fictitious force is the “force” that presses you backwards or forwards in your seat when you’re in a car that suddenly brakes or accelerates. You might think that there’s a mysterious force that suddenly grabs on to your body and pulls it forward when the car brakes; but really, it’s that the car is stopping, and the inertia of your body “wants” to keep moving forward. You experience this leftover inertia as a “force” that pulls you forward.
But gravity’s a real honest-to-Bob force, though, isn’t it? 'Fraid not. Einstein’s big idea was to say that spacetime is “curved”, and that an object left to its own devices will move in space and time in a certain natural way. What’s more, any massive object will cause spacetime to curve; and the resulting “natural” motion in space and time of another object due to this curvature is not to remain the same distance from the first object, but rather to approach it with a certain acceleration. (There are precise, elegant mathematical notions to go along with all this, but I’ll spare you those as well.) The reason we experience gravity as a “force” here on Earth is that we insist on staying the same distance from Earth, deviating from the “natural” path in space and time that our bodies “want” to take — just as when a car brakes, our bodies “want” to keep going, and we experience this as a force pulling us forward in the car.
This is probably much longer of an explanation than you wanted, but hey, what else am I going to do on a sweltering hot Tuesday night?
Hope this helps.