When matter falls into a black hole its energy increases. At the event horizon the energy has increased to infinity. The speed of the matter is c when the energy is infinite.
Energy is frame-dependent, even in classical physics and flat spacetime (for example the total energy of the book in my lap is zero using a frame in which the airliner I'm on is at rest, but not zero using a frame in which the ground 10,000 meters below the airliner is at rest).Matter couldn't possibly have an infinite quantity of energy, could it? I was thinking more along the lines of approaching infinity, but I can't really believe it's possible, just like it can't actually attain lightspeed, right?
Thus, you can make the energy come out to be pretty much whatever you want it to be by choosing your coordinates to produce that particular value (of course energy is still conserved, although the value of the conserved energy is different in different frames). Especially in a curved spacetime, we can find coordinates in which sensible physical quantities such as the energy of an object or the time between ticks of a clock become infinite somewhere - usually this just means that we've made a poor choice of coordinates.
Jartsa's suggestion about speed reaching ##c## and infinite energy at the horizon is an example. The coordinates that work well for observers hovering above the even horizon are Schwarzschild coordinates and just abut everything you'll hear about black holes outside of a serious GR textbook is based on calculations using these coordinates. However, Schwarzchild coordinates have a coordinate singularity at the event horizon, so the infinite energy at the event horizon should not be taken seriously.
It's worth taking the time to study and understand a Kruskal diagram, which draws the spacetime around a black hole using coordinate that do not have a singularty at the event horizon. This allows you to visualize what's really going on as an object falls to and through the horizon.