they became white through evolution! lol. not a good explanation. but ill try and see if i can explain it. ill use rabbits
all rabbits obviously share a common ancestor somewhere down the road. as the rabbits migrated elsewhere they needed to adapt to their environment. so as certain rabbits first got to say a snowy area, the rabbits with the LIGHTEST fur would survive while the rabbits with the darkest fur would not. therefore the rabbits with the lighter fur would reproduce. over time as the lighter furred rabbits kept reproducing, their offspring would have lighter fur than their parents. and as they would get hunted, again the lighter skinned rabbits would survive to reproduce while the darker ones would not. this would continue until the lighter furred ones would keep reproducing and reproducing to eventually get white fur.
so they didnt necessarily start out white, but certainly there was some variation is the rabbits fur which enabled a rabbit with a certain gene to survive and they eventually evolved into white fur