Some are saying that this was a planned leak. Something done to perhaps drum up the hype in the face of a packed E3 and ill thoughts circling Battlefield 4. Whatever it is, you get some new media out of the deal, so just accept it.
Now, about now is where I would embed the video. The problem here is that EA keeps removing it from YouTube. Go figure, right? So instead, all you need to do is download the file and watch it locally. This is probably your best option since EA is pulling this video from YouTube almost as fast as people are uploading it. That, or you get terrible commentary videos from people that feel the need to actually talk over the trailer.
You can get a good look at the single player campaign! I know how much everyone loves those in their Battlefield games.
You can also see bits about the Heist and Hotwire game types. There's some stuff about climbable ropes that can be used anywhere, zip lines (no relation), and more.
Gifs not doing it for you? Don't want to download the video? Alright, then check out some of these webm clips courtesy of NeoGAF.
Alright, now there really can't possibly be any more leaks for this, right? Unless they leak the E3 trailer next or something.