With a diffraction grating, all colors travel at the same speed, but some light waves reflected off/transmitted through the grating end up taking a longer path

With a diffraction grating, all colors travel at the same speed, but some light waves reflected off/transmitted through the grating end up taking a longer path to reach the same point in your eye, and depending on the wavelength, that length difference causes constructive or destructive interference, amplifying some colors and suppressing others. Colors traveling at different speeds through a medium is the usual explanation for color separation through refraction, but it doesn't really help for understanding diffraction.