Which scientist used cepheid variables to measure the distance to faint "nebulas" in our sky, proving they were actually whole other galaxies?

Answer: Edwin Hubble

Cepheids variables are stars of variable luminosity (like a lighthouse), pulsing with specific periods. That is, these are stars that periodically change their brightness (from the perspective of an observer on Earth).

Then, by measuring this period of variation of brightness we can have a measure of the brightness of the star. And knowing this brightness, this can be compared with the star's apparent brightness in the sky to obtain a measure of the distance to the star; thus indicating the distance at which the star is in the Universe.

To understand it better and returning to the comparison with the lighthouse:

If we know how bright is the light of a lighthouse at a specific distance (two meters, for example) and we also know how that brightness changes with distance, as we move away from the lighthouse we will know how much its brightness has been reduced. In this way we can use this measure as a pattern to calculate the luminosity of the object as a function of distance.

Well, this is what Edwin Hubble did to measure the variable brightness of the star Cepheid V1, discovering that the nebula in which it was found was actually a galaxy (Andromeda galaxy), showing that there were more galaxies in the universe and that the Milky Way was not the only one.

Then Hubble continued progressively its measurements with Cepheid variables in more distant galaxies measuring their periods, managing to formulate the famous Hubble law, which would open the way to the knowledge that our universe is expanding.