How did scientists come to the conclusion that it is space that is expanding? How did scientists come to the conclusion that it is space that is expanding and not galaxies moving away from each other as in a giant explosion?
The simplest explanation relies on the statistical unlikelihood of us being at a special place and time in the cosmos.  Let's say, for a moment, that we see galaxies moving away from us because of some cosmic explosion that set them in motion.  In that case, we would expect to see the galaxies in one direction moving faster than the galaxies in another.  Looking towards the center of the explosion, we would see galaxies on the other side moving much faster than if we looked in the other direction.

This, however, is not what we see.  What we observe is all galaxies moving away from us, with a speed proportional to their distance from us.  That means that the further away they are, the faster they are moving.  It doesn't matter what direction we look in; a galaxy 500 Mpc away in one direction will have a recessional velocity approximately the same as one 500 Mpc away in the other direction.

This forces us into one of two conclusions:  either we are at the exact center of the universe, which is statistically impossible, or space itself is expanding, so everything is moving away from everything else.  The animation below shows how this would work.  From any given point in space, it would look like everything is getting further away from you.


Explaining the recession of galaxies due to an explosion also raises many questions:  What caused the explosion?  Why would it start at one particular point?  What were things like before the explosion?  All of these are questions which do not have accepted answers.  Instead, the expansion of space ties directly into the idea of the Big Bang.  The expansion of space itself is what drove the Big Bang.  It is not that the universe is expanding into something.  Instead, space itself is expanding.  However, the topic of the Big Bang is a different subject, so you can look up information on that yourself.