The theory of evolution does not entirely rest on the ancient backs of fossils, but they certainly are an important cornerstone for the theory. No other theory of how all current species of life forms on Earth came to be explains fossils. Also, fossils show that there are far more similarities in life forms than differences. The fossils also suggest that all forms of life like reptiles, fish, birds, animals and people have common ancestors.
This proof that we are more similar than different shows that all species are somehow interrelated. The theory of evolution is that current life forms adapted from older life forms. The original theory stated that this happened in a slow, gradual process, but fossils show that the process went awry and did some very strange experiments at times.
When explaining the theory of evolution, the species usually pointed to as a prime example is that of the horse (equus caballus). We are fortunate that we have a very extensive fossil record of this species. The usual theory is that horses evolved from tiny, muli-toed leaf eaters and gradually grew larger and with less toes.
However, there have been gargantuan sized super horses that have shown up in fossils that throw the gradual growth theory off. There have even been "pygmy horses" (Archeohippus) that were smaller than the traditional first horse (eohippus). But neither of those lines survived to the present day.
Every now and then, you get a foal born with more than four legs. This can happen to foals born of healthy mothers and also foals born with mothers exposed to radiation (such as when the Chernobyl disaster happened). But where do the extra legs branch out? Not from the pelvis but usually from where the extra toes of horse ancestors were. In a modern horse, it looks like the extra legs come from just below the knees or hocks.
Funny enough, there's a famous horse in Nordic mythology that was the steed of the great God Odin. This stallion's name was Sleipnir and he was hard to miss he had eight legs. Often, myths are based on actual persons or animals. Could Sleipnir have been a living fossil? There is a famous photo taken of a foal born (and died) in 1990 after Chernobyl that had eight legs. You can't help but wonder.
The theory of evolution still holds up, even if we didn't find fossils but it would've taken us a lot longer to figure out. Today's genome mapping and DNA research also shows the strong links between all living things.