Recently on the discovery channel I heard that a black hole burps matter out of it when it consumes more than it could process at a time. Given that even light cannot escape from a black hole, how does this "burp" matter escape?
The matter that the black hole is ingesting begins with some initial conditions outside of the black hole's event horizon, and due to numerous processes this surrounding matter loses energy and angular momentum and so falls into the black hole. The physics of this infall process can be quite complicated - especially if an accretion disk forms - but essentially, viscous friction between the particles that compose the infalling matter will cause it to lose momentum, among other possible processes.
There is a theoretical limit of how much matter a black hole (or any astrophysical object) may accrete: the Eddington luminosity limit. Basically, as the black hole swallows the matter it illuminates electromagnetic radiation due to the intense collisions the infalling matter is experiencing near the event horizon. Eddington found that once the luminosity of this emitted radiation reaches a certain limit, the accretor (in this case a black hole) will actually begin to expel the surrounding matter rather than continue swallowing it, due the fact that once the limit is reached the surrounding matter will be blown away by the radiation. See here and here for deeper explanations of this.
So the surrounding matter that is expelled (once the Eddington limit has been reached) did not ever reach the event horizon - it's expelled by radiation before it makes it that close to the black hole.
And here's a lovely image of a double burp! :)
Burp is a misleading word because the explosion energy of compressed hydrogen disk causes the burp energy. The black hole force/energy only draws inwards.
Black hole compression of matter is most powerful energy source in the visible universe.
Accretion disks and quasars happen especially in the early universe when free hydrogen was burped into projectile vomits going travelling nearly at light speed higher than the width of our galaxy.
It's like 1 supernova happening every minute.
Artist impressions and transverse sections of the "burp". https://www.google.fr/search?q=quasar+accretion+disk&num=100&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiF0tnFvpneAhVnyoUKHYkQCMwQ_AUIDigB&biw=1257&bih=646