A standard phrase in popular discussions of the Higgs boson is that "it gives particles mass". To what extent is this a reasonable, pop-science, level of description of the Higgs boson and it's relationship to particles' masses?

Is this phrasing completely misleading?
If not, what would be the next level down in detail to try to explain to someone?

The Higgs field (note it is the field that is important here, not the Higgs boson itself, which is just a ripple in the Higgs field) gives particles mass in the same sense that the strong force gives the proton mass (context: $99\%$ of the mass of the proton comes not from the mass of its constituent quarks, but from the fact that roughly speaking the quarks have a large amount of kinetic energy but are bound by the strong force). If any force confines energy into a small amount of space, then that bound energy has a mass given by $E=mc^2$. This is what the Higgs field does: it binds a massless particle into a small space, and therefore by $E=mc^2$ (and the fact that the particle now has a frame of reference in which it is stationary) that particle has an effective rest mass.  

To get an intuitive feeling for what's going on, as an exercise you can derive $E=mc^2$ by considering a photon confined by a mirror box. The photon is bouncing back and forth exerting pressure on the mirror, and if you try to push the box it will have inertia due to the photon exerting more pressure on the front of the mirror than the back. If you work it out you will find that the mirror box has an effective inertial mass of $m=E/c^2$. The Higgs field provides a force that acts like this mirror box, thereby "giving" mass to the particle inside it.
