The equivalence principle follows naturally from the geometry of general relativity.
The idea is that mass curves space. In this curved space, a 'straight line' is no longer straight. The upshot of this is that if you are standing still, a 'straight line' would actually mean falling down. By inertia, things want to move in straight lines, so we feel gravity. It is an apparent force like the centrifugal force.
If this talk of straight lines in curved space is confusing, consider what passes for a straight line on the surface of a sphere. It still 'curves'.
Can Mach's principle and equivalence principle be derived from some more fundamental laws or are they accepted as axioms?