Though I've more or less come to terms with the math behind the Higgs mechanism, I'm trying to develop a rough intuition for it. Here's how I'm thinking: the Higgs field (like everything) likes to sit at a value which minimizes the potential. For the sombrero potential, at a minimizing value one can identify two directions: the 'angular' one, along which you still have minima of the potential, and the 'radial' one which changes the value of the potential. We expand the Higgs field in that basis, so to speak. The component along the angular direction is the one that corresponds to the massless Goldstone boson, and can be seen via gauge theory to not be physical. So as far as I understand the radial component is what's really the Higgs boson.
Since the field likes to sit at a minimum potential, if you nudge it somehow it'll have a tendency to go back to its resting value; this creates a sort of resistance, like mass is resistance to acceleration. Other fields, like the electron field, interact with the Higgs field; this means nudging the electron field will also nudge the Higgs field, which then provides a resistance, and that makes electrons also have mass.
While it's picturesque it's not very clear to me how the 'resistance to being nudged' gets identified with mass. The Newtonian analogy seems a bit of a stretch, and maybe if we could see it along the lines of $E^2 = p^2 + m^2$ it'd be more convincing.
