ZeroMQ is a publisher / subscriber messaging system. In the case of Bitcoin Core it is a socket which other applications can connect to and get notifications of new events in real time, without having to repeatedly ask the daemon if there are any new events. ZMQ support is useful for any systems interfacing with the network like miners and wallets, which can choose to subscribe either to just transaction notifications or block notifications as their situation requires. It covers a lot of the same use area as the prexisting walletnotify and blocknotify systems, but with significantly improved performance and ease of configuration. These older systems have the unfortunate side effect of spawning a new process for every notification, which tends to incapacitate the host if not used extremely carefully.