The stars Slipher looked at showed unmistakable signs of a Doppler shift, the same mechanism behind that distinctive stretched-out yee-yummm sound cars make as they flash past on a racetrack.
While we may not see that motion directly, if we take spectra of their light, breaking it up into individual narrow colors, we can see the Doppler shift in their spectra.
Star forming gas clouds emit a very narrow wavelength range of radio waves, and this could be used to measure their Doppler shift, how quickly they were moving toward or away from us.