Ok, so here's the deal:
In the Prime Universe the tribbles have been all but extinguished by the Klingons... apart from a couple of tribbles who happened to fall into the hands of the Borg. So the assimilated tribbles create a time vortex and head to the JJ-Verse to have their revenge. The Federation's tribble expert Worf (one of the few who have seen a tribble before during his time on DS9) tries to stop them but eventually just gets dragged into the same time vortex.
After their arrival in the JJVerse the borg tribble ship encounters the USS Fahrenheit under Captain Rob'Oh. The borg tribbles attack and nearly destroy the Fahrenheit. Rob'Oh orders the evacuation of the damaged starship, then sacrificing himself by going on a collision course with the furry Borg sphere.
The rest of the movie sees the destruction of Qo'nos as an act of tribble revenge, Worf meeting his own grandfather (Benedict Cumberbatch as one of those ridgeless Klingons), a lot of lens flares, Sulu wielding an axe, and a couple of Beastie Boys songs.
The movie ends with the surviving tribbles looking at their secret plans to the Tribble Empire’s ultimate weapon, the DEATH FURBALL, an armored space station with enough power to destroy an entire planet.