Edwin Hubble was an astronomer in the 1920s. His research led him to the discovery that the Doppler Effect can be used in space. He noticed that the light from galaxies around us was a low frequency, and was shifted towards the red end of the spectrum (This is known as "The Red Shift"). This proved that those galaxies are moving away from us, so the Universe may be expanding.
So ever since then, that has been the theory. Of course we have just scratched the surface of what we need to know about space, but hopefully in the future we will know even more!