How does gravity keep objects in motion around the sun?

Gravity, per say, doesn't. It's only a player in the game. With only gravity, things would slowly be sucked into the sun, like a black hole. So that's where the other team, inertia comes in. Inertia is Newton's first law of motion, stating that an object in motion will continue to stay in motion unless another outside force messes with it.

And so we have two major forces, gravity and inertia, pushing on one another. Neither win, and as gravity wants to pull into itself and inertia wants to fly away in a straight line, we get a circular path for the object that most call, an orbit.