Simply put, distance.

Gravitational force is proportional to each mass and the reciprocal of the distance between them squared.

So with a black hole, you can get a lot closer to the center of its mass than with a star. If you lose half your distance, you feel 4X the gravity. Since the sun is almost 1.4 million km wide, and a black hole of the same mass would be tens of km wide at most, we can get a lot closer to the black hole. Orbiting 50 km away from a 1-solar-mass black hole (near the event horizon), you would feel almost 200 million times the gravity that you would on the surface of the sun. From the same mass.

The math is more complicated IRL (always is).

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There isn't "more" gravity. Because the mass of the dead star is so dense the gravity has a stronger localised force.

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I've always assumed that a black hole is like a star that has accumulated so much mass that it's gravity has tipped over the point where light can escape. Light no longer leaves it, it bends back.

I am not a physicist either so this is just my opinion. A black hole made from a single star (around at least 8 solar masses) actually should have less mass than the star that created it due to the supernova blowing the outer layers of the progenitor apart. So, unless the star directly collapsed, an unnova, it will be less massive at first. However, close to the singularity will have crazy physics because the mass that collapsed, and we're talking about something like 4 solar masses for the singularity, still has all that gravity but very little "surface area." Think of a female ice skater twirling with her arms outstretched and her hands clenched in fists. With her arms fully extended, she spins slowly. As she pulls her arms in, she spins faster due to conservation of momentum. If you walked into her reach with her arms outstretched you would get punched maybe once a second. If you did the same thing when she had her arms pulled in you could get punched 10 times a second. The strength of the punches hasn't really increased, but the frequency has, so the damage goes up. A black hole is like that ice skater with her arms pulled in: all that gravitational effect is in a much smaller area than before so once you get close enough it all hits you at once.

First, you've gotta toss the point source model for gravity that works best outside gravitic objects - a=µm/r² kinda breaks down at the center of an object where it tries to divide by zero and predicts an insane acceleration right next to the center, while a quick check with first principles will allow you to realise that gravitic acceleration at the center is actually zero because you're surrounded by equal mass at equal distances in all directions so it all cancels out.

See this stackexchange answer for a neat graph of Earth's gravity gradient vs depth

When a large star goes supernova, the explosion sheds the outer layers but compresses the core, which increases the core section's density at and below the boundary layer - which also serves to increase the gravity since you have the same core mass with a much smaller radius.

If that new stronger amount of gravity is sufficient to continue collapsing the core, you get a runaway effect that ends up with a neutron star or black hole.

Smaller stars can't hit those runaway thresholds, and just leaves white or brown dwarves behind (see Chandrasekhar limit) - but if the escape velocity reaches the speed of light, a black hole will pop into existence where the hyper-compressed core once was, containing all its mass, angular momentum, and charge.