A black hole twists space and time around so that time (the future) points inwards towards the singularity. All the possible worldlines of infalling particles go in to the central point and then have nowhere else to go. We don't know what happens next - <PERSON>'s theory doesn't say.
Some think they stop there, forming an object of infinite density. Some think an unknown theory of quantum gravity takes over and some new unknown-to-current-physics effect stops the collapse. The singularity is replaced by some string-theory 'fuzzball' object, or something similar. Some think that they 'fall off the edge of space' and disappear. (Like, if you hold a length of rope at one end and wiggle it, you can get a wave to pass down it to the other end. What 'happens to' the wave when it gets to the end of the rope? Where does it go?) Some exclude the singularity not because they think it necessarily actually disappears but just to keep the maths simple - General Relativity assumes for a lot of its working that spacetime is a 'manifold', which cannot have edges or corners. Some think that they squeeze through the point and emerge into some other domain. (e.g. 'Journey Beyond the Schwarzschild Black Hole Singularity' <PERSON> et al.) Common sense beliefs derived from our ordinary experience of conservation of matter don't necessarily extend to such extreme circumstances. We don't know.
Nor does it really matter, for the purposes of understanding the 'source' of the black hole's gravity, because the singularity is in the future of every particle that has yet to hit it, and causality requires that the future has no effect on the present. 43
The gravity you experience outside a black hole is entirely due to matter falling into the black hole in the distant past, before it reaches the event horizon.
We can draw a picture of the spacetime around a black hole stretched and twisted so that at every point the lightcones point up the page, like in flat <PERSON> space. These are Kruskal-Szekeres coordinates. In this diagram, light always travels at $45^\circ$ angles to the vertical, and local time around any point flows up the page.
(To connect this picture to the point of view of a distant observer seeing 'a spherical black hole floating in space', tilt your head $45^\circ$ to the right. The singularity (thick black line) is the timeline of the point at the centre of the black hole, at radius zero, and the event horizon (dashed lined) is at a fixed radius away from it, which diagrammatic scale distortions have expanded into a 'trumpet' shape centred on the horizon.)
You can see the collapsing star that forms the black hole shaded orange on the left of the diagram. Light from it travels up the page at $45^\circ$ to the vertical. Because of the twisting of spacetime, it moves parallel to the event horizon, stuck on the boundary. It is like water is flowing inwards down a drain, and ripples spread outwards at exactly the speed the water is flowing inwards, hanging on the boundary forever.
Observers hovering outside the black hole, or falling into it, can only see events happening on their past light cone - the two lines pointing downwards at $45^\circ$ to the vertical. When you look towards a black hole, you can see its entire history right back to the beginning, apparently frozen in time - it only looks black because the last few seconds of light emitted are spread out thinly over millions of years.
Likewise, they can only see the same region gravitationally. And like the light, the gravity of the collapsing star is also trapped on/near the event horizon. Even though the matter has long since fallen into the hole and met its fate, the gravitational information 'emitted' from its descent into the hole is still in the process of escaping.
The <PERSON> black hole is a 'vacuum solution', in that it is the shape of empty space surrounding an infinitely dense point mass. It approximates a real black hole, which is the shape of the empty space surrounding a collapsing star. The gravity of the collapsing matter remains behind, trapped forever on the event horizon by the matter's past distortion of spacetime, even long after the star itself has collapsed and fallen into the singularity. 43