They use a method called the doppler effect.
Just as when a train would be passing by you with it`s horn blasting you would hear it change pitches, light does the same thing. If the star is moving away from you the light in the spectrum is shifted to the red end and if it is moving towards you it is shifted to the blue end. The faster it is traveling the farther the shift is. Edwin Hubble discovered that the farther a object is the faster it is moving away from us and by using this information they are able to calculate somewhat the distance to various objects. There are still some questions about what is called the Hubble constant which has to do with the mass of the universe which will give the values for the distance/speed.
Hope this helps.
Clear skies,
Starman