You creatures attack in a certain order every turn. If your opponent runs out of hp before the vampire gets to attack, you do not get the benefit of stealing life.
He's right, but just to clarify:
Vampires only heal you if they attack, as with Empathetic Bonds, Vamp Daggers, Druidic Staffs, and Shards of Gratitude.
For example, if you are at 98 hp and your opponent is at 1, your vamp will only heal you if it was the first creature to attack, or the first one you played (If one dies, the next creature you play just takes it's place in the order) that could deal any damage.
If the first creature you played was Black Dragon, then you wouldn't get any healing, even if the entire rest of your field was Vampires, since they wouldn't attack.