Relativity is complicated, and there isn't a "simple" explanation if you want to throughly understand it (although I would recommend you research special relativity if you really want to understand it).
Our technology is too primitive to accurately test relativity, although all our current tests agree with Einstein's theories of relativity.
I'm not sure if your last question was rhetorical, but I'll answer it anyways. No we did not. In fact, to calculate the power we needed and the gravitational forces we needed to overcome Earth's gravity, we only needed used Newton's equations.
Einstein thought way beyond his time, as with many other theoretical physicists; and his theories, if correct, will prove very useful in the future, assuming that the human race still exists.