Gravity - Simply Explained
Clocks run slower when they are in the presence of gravitational fields. For example, GPS satellites experience time ticking faster by a small factor of 1-in-a-billion. This may not seem significant, but it can cause their position accuracy to be off by about 11km per day!
Rather than asking how gravity slows down time, a more appropriate question would be how time causes gravity. This is because the warping of time actually causes gravity. For instance, the reason why you are held down in your chair is that your lower body is ticking slower than your head. Time flows in a gradient around the Earth, with faster farther from the Earth and slower closer to it, much like the way water in a river flows more quickly in the centre than at the edge.
Imagine two boats floating on a river, one close to the edge and moving slowly, and the other close to the centre and moving more quickly. The slow-moving boat extends an oar which the fast-moving boat grabs onto. The two boats effectively become one object, and the difference in their velocities across the length of the object causes a torque that rotates the overall velocity vector towards the shore.
Similarly, an object in a gravitational field is rotated because of the gradient of time. The object is rotated in the direction of decreasing flow, which in a gravitational field (time) is downwards.