Gravity isn't a Force, So How Does it Move Objects?

But how can this be? Surely, gravity must be a force in order to pull things, right?
To begin answering these questions, you first need to understand that "accelerate" is the proper term, not "pull." The truth is, gravity does not "pull" objects at all; rather, gravity warps spacetime, causing objects to follow the bends that are created and, as a result, they sometimes accelerate.
Meet the Rubber Sheet
To delve into this a bit more, thanks to Albert Einstein's Theories of Relativity, we know that energy tells spacetime how to bend. In this case, mass is generally the most important part of the equation i.e., it is an object's mass energy that bends spacetime.
So in short, mass bends spacetime, and these bends tell energy how to move. In this respect, it is best to think of gravity as the bending of spacetime—like a rubber sheet is warped by a bowling ball, so spacetime is warped by massive objects.
Here, the large ball makes a large bend in spacetime, causing smaller ball to alter its course and follow the dip.Image credit: GSJ
Just like a car travels down a road that has various twists and turns, objects travel along the path of these bends in spacetime. And just like a car speeds up as it goes down a hill, massive objects create extreme bends in spacetime, and gravity is able to accelerate objects as they enter (or approach) deep gravity wells.
As an aside, we call these paths—the path that objects follow through spacetime—a "geodesic."
To better understand how gravity works, and how it is able to accelerate objects, take, for example, the Earth and the Moon. The Earth is a rather massive object. At least, it is when compared with the Moon. As such, our planet causes quite a bend in spacetime. The Moon orbits around our planet because of the warps in spacetime that are caused by the Earth's mass.
So the Moon is just traveling along the bend—the dip, or whatever you want to call it—the downward slope that our planet makes. In this respect, the Moon does not feel any force acting upon it. It it just following a particular path.
But why don't all the asteroids and meteorites that pass our planet fall into orbit?
Ultimately, the path that an object takes depends on a number of factors, such as velocity, trajectory, and the mass of the respective objects. This is why, every day, there are a plethora of comets and asteroids that go shooting past our planet without being captured and thrust into a particular orbit. Likewise, every day, there are a plethora of objects that are captured by Earth and are pulled down to the planet's surface (others may orbit the planet for a time).
Which will happen depends on the aforementioned factors (speed, trajectory, and so on).
If you want to know more about the relationship between gravity and the path that an object will follow, check out this source. It's a good starting point that gets into some of the math behind the aforementioned concepts.