Its important to note that the Higgs field only describes how particles get what is called inertial mass, which is the mass that resists you pushing on it. If you push a truck and a bicycle with the same force, the bicycle moves much faster than the truck. This is due to the truck having more inertial mass which "resists" the push. The Higgs field basically says (in a very hand-waving way) that all particles with mass interact with this field, which bunches up around them in such a way as to them this resistance to pushing or inertial mass.
There is a different type of mass called gravitational mass that says how particles interact with gravity. As far as we can tell, gravitational and inertial masses are equal (this is one of the cornerstone assumptions of general relativity). However, we don't know a good fundamental reason why this must be the case other than that it appears based on our observations.
So all in all, the Higgs field explains only the cause of inertial mass. This appears to be equal to gravitational mass, but we don't know why--and we don't have an explanation for how the Higgs field directly might cause gravitational mass. It will probably require a quantum theory of gravity to explain this.