Hi Steven- let me start with the easy part.
Einstein did not say "energy is actually matter moving at the speed of light". The famous equation E=mc^2 says that energy is the same thing as the thing ('mass') which we multiply by velocity to get momentum, except measured in different units. (There's a part of that mass, called 'rest mass', which appears in a slightly different equation.) If you took the classical kinetic energy formula for mass traveling at speed c, it would give only half that. Einstein (and also I believe Lorentz before him) derived the equation from assuming:
1. The same laws of physics work for any observers in uniform motion with respect to each other.
2. Maxwell's laws of electromagnetism are among the laws of physics.
3. The laws of conservation of energy and conservation of momentum from classical physics can be extended into this new relativistic physics.
The math then needed is actually just some simple algebra. You can find it in any good beginning text on Special Relativity.
Now the other parts of your question are harder, since they're about General Relativity. The key assumption Einstein made in developing GR was that no effect whatsoever of a uniform gravitational field could be 'felt', a slight generalization of Galileo's observation that all objects fall together in such a field. This simple starting point (plus SR, which also is fairly simple) led to the conclusion that our space-time is not like the one described by Euclid and traditional geometry. It isn't 'flat', and in that sense is 'like a blanket'. For example, the ratio of the circumference to the diameter of a circle is not pi, and varies depending on what's in or near the circle. You can get the same thing on a bulge in a blanket, if you measure the diameter along the blanket. The math used is differential geometry and tensor calculus, both of which Einstein needed to get help with. David Hilbert was one of the key helpers.
I better stop here before going too far beyond the part where I know what I'm talking about.
(published on 10/22/2007)