How gravity works in a black hole?

Gravity works exactly the same around a black hole as it does everywhere else in the universe:

The mass of the black hole distorts spacetime, and then objects follow the shortest route in the curved spacetime. The only difference between the gravity of a black-hole and the gravity of Earth is that the Black hole has a much larger mass in a much smaller volume, so the gravitational field is much stronger.

The strength of gravity has several consequences I'll mention three:

First close to the black hole spacetime is so curved that light can't escape. Now all gravity bends light a bit, but you usually don't notice it. A black hole bends light a lot. The edge of the black hole is called an event horizon and it is the closest distance that light can go to a black hole and not fall in. Nothing can ever pass from within the event horizon.

Next there are tidal forces. All gravity can cause tides: the force of the earth gravity is a bit more on your feet than on your head. Usually you don't notice. However a solar mass black hole has such extreme tides that you would be pulled apart by the difference in force (this is the spaghettification you mention)

Finally, inside the black hole, space works like time. You can't change your direction in time (unless you have a Delorean) Inside a black hole you can't change direction in space.

But remember the only difference between the gravity around a black hole and the Earth is a matter of degree.