How does the Higgs field enable masses to attract each other?

The Higgs field mediates so-called Yukawa interactions between fermions. These interactions are of the form $$ y H \bar Q d_R + \text{h.c} $$ and result in masses by the Higgs mechanism. The Higgs field $H$ acquires a vacuum expectation value (VEV) because of the Mexican hat shape of the Higgs potential, $H \to \frac1{\sqrt 2}(0, h + v)$. This results in masses $$ \frac1{\sqrt 2} y v \bar{q} q = m \bar{q} q $$ and interactions of the form $$ \frac1{\sqrt 2} y h \bar{q} q = \frac{m}{\sqrt 2 v} h \bar{q} q $$ The Yukawa interactions generate an attractive potential between fermions exchanging Higgs bosons. Thus, the Higgs field enables fermions to attract each other via the same Yukawa interaction that generates mass, and the strength of the attraction is proportional to mass.

Remember, though, that the Higgs field and Yukawa interactions have nothing to do with gravity.

Good question! Although the Higgs field gives rise to the masses of fundamental particles, it is not responsible for the gravitational force between massive particles. In other words, due to spontaneous symmetry breaking, the Higgs field causes mass terms in the Lagrangians for the effective field theories of the fermions in the standard model. However, the resulting effective field theories do not describe the gravitational force. Somehow, gravity then uses these masses to produce an attraction (actually it uses these masses to curve spacetime according to general relativity). This gravitational mechanism is not currently incorporated into a unified theory that involves all the forces of nature.

Based on the comments, it seems that I may have misunderstood the question of the OP. So here, I'll try to give a better picture:

Above the high energy level where the electroweak symmetry breaking occurs (electroweak scale), the Higgs field (more than just the Higgs boson) exists as a scalar field that couples to the fermion fields via Yukawa couplings.

Below the electroweak scale, the Higgs field gives rise to a vacuum expectation value (VEV), which turns the Yukawa coupling terms into mass terms in the Lagrangian. The only remaining Yukawa coupling terms is the one with the Higgs boson, which itself also acquires a mass.

Now to clear up some misconception. If we ignore gravity (as the OP seems to want to do) then there is no force between masses. In other words, the Yukawa coupling term that survives beneath the electroweak scale cannot be interpreted as an interaction for which the mass represents the coupling strength. The reason is simple, consider for instance the electromagnetic interaction: its interaction term in the Lagragian contains a coupling constant namely the electric charge that tells us how strong the force is. By contrast, the coupling constant associated with the Yukawa coupling with the Higgs boson is not the mass of the. In fact the mass terms and the Yukawa coupling terms are different terms in the Lagragian. It is true that the Yukawa couplings help to determine the values of the different masses, but that does not mean that the masses play a role in the interation.

It should also be mentioned that the remaining Yukawa coupling gives rise to an extremely weak force, due to the large mass of the Higgs boson; weaker even than the weak nuclear force, which gives us beta decay. It is effectively a contact force that one can only really observe a very high energies.

In summary, one should not confuse the Yukawa coupling associated with the Higgs boson as a force between masses (such as gravity).

There are a few points we need to get clear from the outset: the Higgs boson is not responsible for giving particles mass nor for the gravitational attraction between them. The Higgs field is responsible for giving particles mass, and the boson recently discovered at the LHC emerges as a side effect of this. I rant about this in my answer to How does the Higgs Boson gain mass itself? innisfree's answer gives a nice summary of the Higgs mechanism and I won't attempt to improve on that. A fuller but still accessible explanation is given on Matt Strassler's blog and you should take the time to read this is you are interested to find out what is really going on.

A side note: virtual Higgs exchange does actually lead to a short range Yukawa force, though whether this really counts as a force is debatable. This is explored in the answers to Why isn't Higgs coupling considered a fifth fundamental force?

But back to your question. The Standard Model is formulated in flat space, and the mass that the Higgs creates is the inertial mass. Gravity is not included in the Standard Model so it does not and cannot explain how the inertial mass generated by the Higgs mechanism causes a gravitational force. To ask how the Higgs mechanism causes spacetime curvature is therefore a meaningless question. To address the issue we would need some theory of quantum gravity, and currently no complete theory of quantum gravity exists.

You mention gravitons in your question. If we attempt to quantise gravity using quantum field theory in an obvious way we get a theory where the gravitational force is transmitted by virtual gravitons (real gravitons are responsible for gravitational waves). But there isn't a simple way to understand how this is equivalent to a curved spacetime. If we do the maths then we get the same results as a curved spacetime approach, but I'm afraid the maths involved is rather opaque.

And finally, one key point often ignored is that it isn't just mass that creates spacetime curvature i.e. gravity. GR treats mass and energy as the same thing and related by Einstein's famous equation $E=mc^2$. So as far as GR is concerned it doesn't care whether the Higgs mechanism has given particles mass or not. It's the energy of the particle that counts. Well, that's not quite true because massless particles travel at the speed of light and calculating the gravitational force between objects travelling at the speed of light is somewhat involved. However the basic principle remains. In fact a lot of the mass of particles like protons is due to massless gluons within them and not to the Higgs mechanism at all.