There is no answer to this question. Relativity simply allows one to calculate how much curvature is created in a particular mass configuration, and what the resulting gravitational forces are. It doesn't try to explain why the curvature occurs.
I suppose it's possible that mass interacting with the higgs field could give rise to motion in the way an object will be pushed from a high density area of liquid to a low density area as mentioned in another comment here..
Those answers are so vague and abstract and far from a conceptual understanding from most people that have them told to me, that one comes to wonder how much they resemble the mystic explanations of the universe by priests and samans of yore in functionality.
E.g. a lot of big words that many repeat but few are supposed to understand (and even those, not that good).
What is pulling it then?