I actually don't think ZKPs solve the age verification problem (for a variety of reasons that won't fit here), but they solve plenty of more interesting problems. Private transactions in Zcash, for example.
If they had no clue who it was, like the typical http web, they could not allow a sender to be a prover, since they would not be able to verify their balance. At some point a balance inquiry would have to reveal who they are. It only works if you trust the banker.
Or by being a middleman. You send [payment service] the money and they send it to the recipient, hiding your info.