Wolves eat the white bunnies as they are easily spotted
During mating season one of the brown rabbits with the dominant fur allele, mates and reproduces with a rabbit with the recessive fur allele of white fur. When they reproduce they know that only some of they're offspring will survive to reproduce so they have more offspring.
The two rabbits had a total of eight kittens, and with variation in the genetic profiles of the kittens. Two of them got the white fur allele and the other 6 had the brown fur allele.
Survival of the Fittest
Since the bunnies live in a forest the white bunnies are not easily hidden and stick out. Since the bunnies natural predators are wolves this makes the white bunnies easy to spot and therefore easy prey. Only a few white bunnies survive to reproduction.
The White Bunnies Thrive in the new Environment
A harsh winter comes around and blankets everything in white. This is where the tides turn as the white bunnies can blend in with the surrounding environment easier than the dark, brown bunnies. This results in more white bunnies reproducing and passing on the favored white fur gene.
Since there are more white bunnies they are able to reproduce and pass on the white fur allele. Of course the dominant fur allele is still there but it becomes less common as the white bunnies are able to survive better. As this results in less wolves eating them.
Since the white bunnies are harder to find in the snow, they're natural predators the wolves cannot find them as well and this gives them an advantage over the brown colored bunnies. Now the white bunny population grows as more white bunnies live to reproduction.