Astronomers measure the movement of objects relative to us using Doppler shift. When you hear a train coming, its whistle is heard at a different frequency compared to when it is receding, right? In the same way, light also has a Doppler shift, whereby its frequency is shifted depending on the motion of the emitting object.
Astronomers observed that light from distant objects in the universe is redshifted (shift in the frequency of light towards red color), which tells us that the objects are all receding away from us. This is true in whatever direction you look at: all the distant galaxies are going away from us. This can only be due to the fact that the Universe is expanding.
Further, by measuring the distance to the galaxies, one finds that the velocity of recession is proportional to the distance of the galaxy from us. This is called Hubble law after Edwin Hubble who was the first to discover it.
This page was last updated June 27, 2015.