Q:

How do we know that clocks slow down relative to each other?

For example if a body in motion experiences time dilation, why does it see a body at rest slow down relative to it? wouldn't it make more sense if the body at rest has a faster clock relative to it?

A:

No, since by the principle of relativity: A body in constant velocity motion cannot determine whether it is in motion in a certain direction or whether everything else is in motion in the other direction. No physical experiment can determine this hence for all purposes a body in motion will simply claim that it is at rest while the other body is in motion. 
Hence from Special Relativity, it will measure the other bodies clocks(which are in motion) as ticking at a slower rate.

