I am a very beginner when it comes to the Higgs mechanism, so the best my brain could understand (without the gory math) is the fact that the Higgs field causes fermions to flip between their left- and right-handed version, which causes them to become massive.
Most explanations end here, leaving only more questions. I don't get it why this constant changing between left and right spin states forces the particle to move slower than c and behave like a massive one.
I'm afraid you cannot avoid the math and get a mere "story" with deeply misleading and unsound logical paths of causation... Math can be tricky, but "the book of nature is written in math", as Galileo said. He did not exaggerate.
Chirality is a Lorentz-invariant, mathematical, unintuitive notion: a Dirac fermion ψ, has a L- and a R-chiral piece. Chirality is defined through the operator γ5, which has eigenvalues ±1. Any Dirac field can thus be projected into its Left- or Right-chiral component by acting with the projection operators ½(1 − γ5) or ½(1 + γ5), respectively, on ψ. Books describe how chirality is connected to helicity and spin, but you might not need to know the details, if the math cloys you.
What you need to know is that as fermions propagate, they maintain their chirality, unless they have a mass, which allows them to flip it. A mass term, $m\overline{\psi_L} \psi_R$ + hermitian conjugate, is a 'bridge' between L and R components of the fermion; it allows L and R to "leak" into each other. If you are a fermion, you can't have a mass without this term.
In the Standard Model, L fermions have dramatically different gauge symmetry properties; contrasted to R fermions, which have no such symmetry. So, to have a symmetric theory, you must couple the L fermions with the Higgs field, which also has such a (gauge) symmetry. You then tack on a R fermion to this gauge invariant term, and you end up with a ("Yukawa") term: a term like the mass term above, with a Higgs field in place of the mass.
Through an elaborate group theoretical mechanism, (SSB, not the Higgs mechanism that only concerns gauge boson masses) the Higgs field picks up a constant term in the bizarre just-so vacuum of the Standard Model, and so it nets a term that is indistinguishable from the mass term I just wrote.
This is what happens. I'm not sure how you want things to cause other things. They just fit together, and in no other way, like a sudoku puzzle. Seeking a "cause" might not be helpful in understanding how the machinery works. The odd popular science "story" of various massless fermions and bosons plowing through a jar of "molasses" Higgs fields and thereby "slowing down from the speed of light" is an alarmingly gonzo metaphor; truly confusing and an invitation to many more questions than can be answered... All you need to understand is what the pieces are, what the couplings of bosons and fermions to the Higgs field are (the Yukawa ones are numerous and mysteriously inexplicable) and what actually happens. Massive particles can never achieve the speed of light, but their speed depends on the connection of their momentum to their energy, which depends on the mass, by Lorentz invariance.