The title is misleading. They didn't "regrow" the man's finger, they merely attached it back when it was in a bad condition.
Attaching back a recently-severed limb is possible, because the body can identify it and "reconnect" it to the body. It can be done even with foreign body parts; This is what allows transplants.
But growing a body part back like a lizard is still not possible (yet). Adult human cells simply don't have the capability to do it. Perhaps stem-cell research could provide solutions to that problem, since as you know embryonic cells have certain characteristics required for human development that adult cells do not.