I'm not sure what you're asking, BdS. What kind of physical mechanism are you looking for?
General Relativity is a theory that describes (among other things) why things with mass attract one another gravitationally. It describes that attraction as resulting from a curvature, or warping, of a spacetime manifold. GR is a theory, and the idea of a spacetime manifold is just that: an idea, or concept, used to explain and accurately predict observed phenomena.
Depending on what you mean, then, there are a few possible answers. Answer number 1 is that that General Relativity itself is the required "mechanism".
Answer number 2 is that some things in our universe have this property called "mass", and one of the functions of mass is to warp spacetime, so in that sense, matter itself is the "mechanism" - wherever there is matter, there will also be warped spacetime.
Answer number 3 would be whatever the answer is to the follow-up question "Yes, but why does matter warp spacetime?" This is where science parts company with philosophy. Science just says something like "We don't need to know that. It just does. That's what we observe that it does." Philosophy, on the other hand, goes down the rabbit hole, asking "Why is there mass and spacetime in the first place?" and "Why is there something rather than nothing?" and similar ontological questions. There are various ideas.