Gravity doesn't actually pull you "down." You are locked in a tug-of-war with every speck of dust and chunk of rock on Earth, being yanked in billions of directions at once.
Every atom in the planet exerts a tiny, invisible pull on your body. Because the Earth is a sphere, the rock thousands of miles to your left is pulling you to the left, and the rock thousands of miles to your right is pulling you to the right. The pull from the northern hemisphere cancels out the pull from the southern hemisphere. The only direction that doesn't get canceled out is the straight path toward the center of the planet.
In 1687, Isaac Newton proved this mathematically with what is now called the Shell Theorem. He showed that for any uniform sphere of matter, all the individual gravitational pulls from every particle add up perfectly to simulate a single point of mass at the exact center. As far as physics is concerned, the gravitational pull you feel on the surface is exactly the same as if all of Earth's mass were concentrated into a microscopic dot at the core.
This geometry means "down" is not a fixed direction in space, but simply a local vector pointing toward the center of mass. A person standing in Australia and a person standing in Canada are both pointing their feet toward the same central point without either of them "falling off."
Gravity is also the reason planets are spherical in the first place. Once a rocky body in space reaches a certain mass—usually about 400 miles across—its gravitational pull becomes so strong that it crushes its own material inward. The solid rock behaves like a slow-moving liquid, collapsing under its own weight until all matter is as close to the center of gravity as geometrically possible.