Hubble's Law of cosmological expansion was first formulated by Edwin Hubble in 1929.
Hubble compared the distances to galaxies to their redshift and found a linear relationship. He interpreted the redshift as being caused by the receding velocity of the galaxies.
The recession-distance relationship is interpreted as an overall expansion of the Universe. As light travels towards us from the distant galaxies, it is stretched over time by the ever expanding space it is travelling through. The longer it travels, the more the wavelengths are increased (reddened).
This shift is different than the doppler effect, which is caused by the motion of the light source itself. It is similar to drawing an image on a piece of rubber or latex and then distorting the image by stretching.