It's a living drug because it's a drug which is in some sense alive. It's made up of T-cells, from the patient's body, but modified.
We also distinguish "live vaccine" from "inactivated" or "killed" vaccines, for example OPV the cheapest way to prevent polio is a live vaccine, it's a version of the Polio virus that has adapted to a slow life in a cold medium, and then you eat it (literally, it was traditionally given by adding one drop to a sugar cube, the patient eats the sugar cube, delicious) and your gut goes "What? This is alien - kill it" and you become immune to Polio before it spins back up and causes any harm.
Live vaccines are dangerous (which is why we don't use OPV in countries with no polio and plenty of money, even though it is more effective than any alternative) and no doubt this drug is also dangerous, but hey, it's for people with a terminal disease so what's the worst that could happen?