How did scientists determine the speed of the expansion of the universe?
By looking at how fast other galaxies are moving away from us.
When we look at other galaxies in the universe we can see that they are moving away from us, at least most of them. The ones that are moving away experience redshift from our point of view. This means that the light coming from them increases in wavelength and appears redder than the light coming from galaxies that are moving towards us. This information is used to determine the speed of the expansion.