The earth has lots of mass, and therefore lots of gravity, so it bends space and time enough to be measured.
Gravitational time dilation occurs whenever there is difference in the strength of gravity, no matter how small that difference is. Time itself is slowing down and speeding up because of the relativistic way in which mass warps space and time. 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.
It is only relative to a reference frame with weaker gravity that his time runs slow. We should note here, however, that an observer in the strong gravity experiences his time as running normal. The stronger the gravity, the more spacetime curves, and the slower time itself proceeds. The gravitational field is really a curving of space and time. Gravitational time dilation occurs because objects with a lot of mass create a strong gravitational field. It is predicted by Einstein's theory of General Relativity and has by verified multiple times by experiments. This effect is known as "gravitational time dilation". Yes, time goes faster the farther away you are from the earth's surface compared to the time on the surface of the earth. Although this is a very weak effect, the time difference can be measured on the scale of meters using atomic clocks. Earth's mass warps space and time so that time actually runs slower the closer you are to earth's surface.