Fossils don’t live for billions of years. The organism that eventually gives rise to a fossil will have lived out its normal life. Upon death, if its body is in the right place, and assuming the dead body is not scavenged upon by some other creature, the remains of that creature can become fossilised, and it is the fossil that can survive hundreds of millions of years. I’m sure a great many fossils have been destroyed in subsequent geological disturbances, such as new volcanoes, but occasionally deep layers of rock containing fossilised remains are thrust to the surface where you can very easily see them.
For for information on fossils, have a read of this: