"Aww, that PlayStation 4 has a little brother! What does it do?"
This actual real world quote is one of several heard when PlayStation 4 owners were first exposed to PlayStation VR at a local demo station. When you first see everything needed to connect your PlayStation VR to the PlayStation 4, this extra box that has to sit next to your console stands out a bit. Sony calls it the Processing Unit, and it's a critical component in making sure your PlaySyation 4 is VR ready.
Why do you need this extra box for PlayStation VR? Glad you asked!
Your PlayStation 4 doesn't do anything with spatial audio out of the box, which makes sense when you consider its original purpose was to play games on a television that doesn't move. Spatial audio is tricky, using the movement of your head to determine where audio in the game should be coming from. This Processing Unit handles the processing necessary to make that spatial audio happen, so when you're playing a game on your PSVR and something tries to sneak up behind you those footsteps can be heard as though they are behind you. Cool, right?
In a perfect world this would all be baked into the next PlayStation release.
As you may have noticed while looking at the back of your PlayStation 4, you only have one HDMI port. That port is used to display everything happening in your PS4 over on the television. It'd be terribly inconvenient to have to reach around to the back of your entertainment center to disconnect that HDMI cable every time you wanted to connect a PlayStation, so this extra box offers up a spare HDMI port. Not only is it more convenient from a cable management perspective, but it ensures you can now have your game displayed on both the PlayStation VR and PlayStation 4 at the same time. This is important if you want to share your VR experience with others in the room, but it's also a big deal for local multiplayer games where only one person is in VR.
Sony is also leaning on this box for Cinematic Mode, when you decide you want to play normal PS4 games in VR or you want to kick back and watch a movie. In these situations, the VR users are seeing more than what is normally shown on the television. A virtual environment like a theater is drawn so the big screen you see isn't just surrounded by black nothingness, and that extra information isn't useful to show on the television.
While it does mean you've got a number of extra cables and an extra box to sort out on your entertainment center when the PSVR arrives — and so far Sony has not made any mention of a white version of the box to match your console if you don't own a standard black PS4 — having the box means you get a much better overall experience. In a perfect world this would all be baked into the next PlayStation release, so keep your fingers crossed!