How do the scientists know the universe is expanding?

Answer 1

The main piece of evidence (there are others) is the redshift from distant galaxies. This is usually interpreted as a result of the Doppler effect - i.e., the galaxies are moving away from us.

Answer 2

The expansion of the universe can be measured through red shift, a distal Doppler shift that causes the light emitted by or reflect off of an object to decrease in frequency. Because we know, through spectroscopy, what the emission spectra of the elements that make up a star should be, we are able to determine how fast a star is moving toward or away from us by comparing how significant the Doppler shift of the star's emission spectrum is. The first to observe red shift in the universe was Sir Edwin Hubble, who actually believed the universe to be static. His findings, however, were the first evidence for Georges Lemaître's theory of expansion of the universe.

In the modern day, one can make rudimentary calculations for the expansion of the universe using fairly simple equipment in one's own back yard. Simply point a telescope at a galaxy, any galaxy but the Milky Way (since we're in it). Direct the light from the telescope through a prism, and either record the resulting spectrum or expose some photo paper to it. Compare this spectrum to the emission spectra of various elements. This will allow you to determine not only Doppler shift, but the composition of the stars you are looking at. If desired, you can perform the same experiment on several other galaxies. You will not get results anything like so accurate or detailed as those of the Hubble telescope's imagery, but proving that the universe is expanding is simply a matter of gazing up into a starry night.