Thanks to a huge rise in tablet and smartphone gaming, handhelds like the 3DS and the PS Vita have seen terrible sales. And even when Sony lowered the pricing for the Vita and sales increased, it still wasn't up to Sony's standards, and they haven't given up on the portable console yet.
For an incentive to those already interested in both the PS4 and the PS Vita, Sony are making it possible to remotely play PlayStation 4 games straight on your PS Vita. It's most likely that the game will be streamed from the PS4 and will work in a similar way to the Wii U's gamepad.
However, no games will need this extra functionality, and it really just serves as a portable copy of your game, so if you're ever fall into the predicament of who gets to use the TV first, you can just switch on your PlayStation, power up your PS Vita, and then change the channel back to live television once you're set up on the handheld.
Sony has stated on their official website that "We would like every PS4 game to be able to use this service as we are sure they will, in time." So it sounds like not all games will support the cross-Vita functionality, however it is accompanied of a nice picture of Killzone Shadow Fall being played on the PS Vita, so there's no doubt that big games are getting the remote support.