How do black holes allow time travel to the past?

They don't.

Real black holes are "regions of space in which the gravitational field is so powerful that nothing, not even light, can escape its pull after having fallen past its event horizon".

Black holes are also singularities in which everything is crushed by the gravitational fields into a region with zero volume. This is an infinitely dense region, and as such, everything there has zero width, length and height.

In short, a black hole will not allow time travel to the past.
They (essentially) vapourize everything. Don't believe every sci-fi book you read. But be my guest, try it!

NO

Hi. Assume that a traveler has approached an event horizon in a ship that shields against tidal forces. The traveler will experience a time dilation that will move her FORWARD in time as far as an outside observer is concerned. (Time slows near an event horizon due to acceleration to near light speed.) When she finally breaks free of the event horizon she will have experienced a shift towards the future, not back in time. So, No.

You might be interested in the following links:

http://science.howstuffworks.com/time-travel.htm
http://en.wikipedia.org/wiki/Time_travel

To be precise, a black hole would likely not provide a mechanism for travel into the past. However, there are related objects that might allow for such travel.
For instance, wormholes, and perhaps naked singularities might allow for travel into the past (provided that such things really exist).

Einstein's general theory of relativity does not, on its own, rule out travel into the past, and in fact there are solutions to general relativity that allow for it to happen. The big question is, "Even if Einstein's mathematical theory allows for those kinds of solutions, does that automatically mean that those objects exist in the physical universe?".
Many (if not most) physicists believe that some extra rule must be taken into account in addition to general relativity. For instance, Professor Stephen Hawking's "Chronology Protection Conjecture" would prevent the conditions for travel into the past. It remains to be seen whether travel or communication into the past is possible, so Hawking's conjecture will (for now) remain just that --- a conjecture.

I must point out that the notion of time travel into the past is not just a crackpot idea that physicists do not take seriously. There have been many legitimate theoretical studies of those solutions of general relativity that allow for travel into the past. A physicist always wants to hedge his bets, so that (even if he thinks time travel is unlikely) the study of the properties of time travel is important, just in case it is someday discovered that time travel really *is* possible. Thus, a physicist could attempt to determine if paradoxes can happen (like a person killing their earlier self), or if general relativity and quantum theory somehow smooth things out so that paradoxes cannot happen. Or a physicist could attempt to discover (mathematically, that is) what the properties of the matter and energy must be in order to make time travel possible. Once that is known then particle physicists would know what to look for in the large accelerator experiments. And there are other example questions that a physicist might explore.

I also want to point out that there are different kinds of black holes (e.g. Schwarzschild black holes, Kerr black holes, Reissner-Nordstrom black holes, Kerr-Newmann black holes, higher dimensional black holes, etc.).
The discoverer of the mathematical solution of general relativity that describes an uncharged and rotating black hole (Roy Kerr), discovered that the singularity inside the Kerr black hole is a ring, not a point. He believed that for a very large and rapidly rotating black hole it might be possible to fly through the ring without hitting the singularity, and possibly emerge elsewhere in the universe, or elsewhen. His solution also, technically speaking, allows for the spin of the black hole to be so high that the ring singularity is no longer hidden from the rest of the universe behind an event horizon or any apparent horizons. This is called a naked singularity, and if it exists then it could cause some interesting time-travel-like effects (effects preceding their own causes, for example).

More on these ideas can be found on the web, starting with the links that I provided above.