Hopefully at your level
Newton's theory may not be true, but it has to be at least a good approximation. If it wasn't then it wouldn't have stood up to observation for so many centuries.
Therefore any viable replacement must explain not only previous facts, but also why Newton's theory seemed to hold.
In the case of Einstein's theory, in the case of weak gravitational fields and things moving much slower than the speed of light, most of the factors in the theory become very small and drop out. What is left over is Newton's theory.
And, of course, the measurements which tested Newton's theory were things in our Solar System, which are in relatively small gravitational fields, and which move slowly (compared to light). An example of how close Newton's theories are is that the orbit of Mercury turns out to precess an extra 43 arc-seconds a century. That is a third of a percent of the way around a circle. In a century.
Just to give a sense of how science has to work, even decades after there was apparently general consensus on the measurement and its meaning, it was still open to further clarification. See [link|http://math.ucr.edu/home/baez/physics/Relativity/GR/mercury_orbit.html|http://math.ucr.edu/...ercury_orbit.html] which talks about the measurement of the Sun's oblateness and how that might affect this result.
Cheers,
Ben
