The theory of expansion of the Universe is supported by evidence that shows that the Universe is expanding, in all directions.
For example, when viewed from Earth, galaxies in all the directions we can observe are moving away from us - this is the cause of the red shift phenomenon - and those farther away are moving the fastest.
The action of gravity works against expansion,
which means that, if the density of the Universe is great enough, eventually it will overcome the expansion and everything will be pulled back in a "big crunch".
Alternatively, if the density is small enough, expansion will never stop. There is a critical value for density at which the expansion will continue forever, at a gradually-decreasing rate.
Peculiarly enough, current calculations based on evidence from, for example, the Hubble Telescope, show a value very close to the critical density.