How does it give elementary particles mass?
Watching that video earlier should be more helpful than listening to my possibly inaccurate explanations. But if you still want to listen anyways...
From what I understand, elementary particles gain mass by interacting with the Higgs field. Higgs bosons, like photons and gluons, are carriers of one of the fundamental forces that form fields that permeate space, like the electric field. Force carrier particles are in fact just oscillations and disturbances in these fields (yes, you sense a disturbance in the Force); as such the Higgs bosons are oscillations in a field called the Higgs field. The Higgs field is the important part; the Higgs boson is only an indicator of the field's existence or something like that. Some elementary particles move more easily through the Higgs field, and these particles have less mass as a result. Why? I don't really know. Mass is a difficult thing to define exactly, because once you get down to quantum levels the definitions of physical quantities like mass and momentum become rather abstract and detached from everyday life; I think it eventually gets down to quantum numbers or something.
I probably got a number of things wrong in this post, but my physics professor isn't here to correct me. I haven't even started my second year of university, after all. Believe at your own risk.