How and why gravity affect time

Yes, time goes faster the farther away you are from the earth's surface compared to the time on the surface of the earth. This effect is known as "gravitational time dilation". It is predicted by Einstein's theory of General Relativity and has by verified multiple times by experiments. Gravitational time dilation occurs because objects with a lot of mass create a strong gravitational field. The gravitational field is really a curving of space and time. The stronger the gravity, the more spacetime curves, and the slower time itself proceeds. We should note here, however, that an observer in the strong gravity experiences his time as running normal. It is only relative to a reference frame with weaker gravity that his time runs slow. A person in strong gravity therefore sees his clock run normal and sees the clock in weak gravity run fast, while the person in weak gravity sees his clock run normal and the other clock run slow. There is nothing wrong with the clocks. Time itself is slowing down and speeding up because of the relativistic way in which mass warps space and time.