They don’t. Mass curves space towards it. The more mass there is, the more it curves, and the closer you are to it, the more it curves. A ball on a sheet has those properties: a bigger ball bends the sheet more, and the sheet is bent more closer to the ball. That’s about where the similarities end. It’s a useful introductory metaphor, but it doesn’t work as a model of physics, even if you scale it up to three dimensions.
As for how massive objects bend space at all, the best answer we have right now is “they just do.” As far as we know, that’s just how the universe works. If there’s mass (or energy), space gets bent.
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