Why we multiply path difference by refractive index

The optical path length is actually a description of the time the light ray needs to pass through a system of optical elements. To get the time of a specific piece of the path you divide with the light speed.

t=s/c

Now for a medium with a different refractive index than 1 we have

c=c0/n (not c0*n, its just defined that way)

Using that we get

t=n*s/c0

Now we just kill the vacuum light speed c0 from all the terms that add up to make the optical time and get the optical path, where every piece is multiplied with the corresponding refractive index.