The PS Vita TV works by connecting to the Internet, a TV display, and a PlayStation 4 console simultaneously. If someone is watching a movie on the TV hooked up to a PS4, turn on the PS4, sync the PS Vita TV to another display, and you'll be able to stream PS4 games from the console to the other TV screen.
It also acts as an ad hoc gaming extension for the PS Vita portable console. You can use the PS Vita TV to enable multiplayer gaming with other PS Vita users in the room. Games that use the PS Vita's touchscreen capabilities will not be compatible with the PS Vita TV, unfortunately.