How exactly does time slow down near a black hole?

This web page provides a good explanation.

To oversimplify the explanation, you have to understand the curvature of space time around a black hole. The basic principle is that because of the curvature of spacetime around a black hole, the amount of "distance" a beam of light has to cover is greater near a black hole. However, to an observer in that gravitational field, light must appear to always be 300,000 km/sec, time has to slow down for that individual as compared to someone outside that gravitational field as related by the time/distance relationship of speed.

Or as the web page says:

If acceleration is equivalent to gravitation, it follows that the predictions of Special Relativity must also be valid for very strong gravitational fields. The curvature of spacetime by matter therefore not only stretches or shrinks distances, depending on their direction with respect to the gravitational field, but also appears to slow down the flow of time. This effect is called gravitational time dilation. In most circumstances, such gravitational time dilation is minuscule and hardly observable, but it can become very significant when spacetime is curved by a massive object, such as a black hole.

A black hole is the most compact matter imaginable. It is an extremely massive and dense object in space that is thought to be formed by a star collapsing under its own gravity. Black holes are black, because nothing, not even light, can escape from its extreme gravity. The existence of black holes is not yet firmly established. Major advances in computation are only now enabling scientists to simulate how black holes form, evolve, and interact. They are betting on powerful instruments now under construction to confirm that these exotic objects actually exist.

This web page provides a large series of links for further research into the subject: http://casa.colorado.edu/~ajsh/relativity.html

A good analogy for the strangeness of space and time around a black hole is traveling from the US to Canada. You feel about the same and the surroundings look the same (like nothing special happening to you when you cross the black hole's event horizon), and the prices in the stores look about the same, but if you try to use the money you brought with you, you suddenly have to make these non-local corrections. Likewise, your own personal time always "feels" the same when you explore a black hole, but your clock runs slower than someone else's clock that is farther from the black hole.

And in fact, crossing the black hole's event horizon is the equivalent of changing your money over to (worthless!!!!) Zimbabwean dollars- your clock seems to stop entirely, from the point of view of someone far from the black hole, even though things seem just fine from your own point of view.

PS- A black hole can be used only for time travel into the future! Just hang out close to the event horizon for a while and then return. Much more time may have passed for everyone else because your clock seemed to run so slowly.

Time slows down near any massive body; black holes are merely the most extreme example. GPS satellites orbiting the Earth have to correct for the fact that time passes very very slightly more slowly on the Earth's surface than it does in geosynchronous orbit -- by about one second per every 60 years.

In a sense, gravity and time dilation are the same thing: they are both consequences of the curvature of spacetime near a massive body. You can't have one without the other.