It doesn't, it's just a lot denser. A star has a lot of inner pressure from the ongoing fusions inside of it. When these stop or reduce, the star collapses from its own gravity, as soon as its radius is smaller than the event horizon (i.e. the radius at which the escape velocity exceeds the speed of light) you have a black hole.
That's basically what I keep seeing in the documentaries, but it's still not sinking in. For instance, being denser I could see it making a 'deeper' 'dimple' in the rubber membrane demos they use...but it would also be a much smaller diameter dimple, so only things really close and really small could 'fall' into it ?
