If I'm understanding, you're asking: How does the Higgs field "give" particles mass?
This is an explanation given by Henry of MinutePhysics
(To be clear,
we're talking about the Higgs field and NOT the Higgs Boson, which is merely an excitation
leftover after the process we're about to explain. But I digress…back to mass!)
To begin, we need to know what we even mean by "mass" - so we'll head the other direction and talk about what it means to be massless: This may sound crazy, but the defining feature of any particle without mass is that it travels at the speed of light. In fact, if we're honest it should really be called the "speed of massless particles," but since the first massless particles we knew about were photons of light, the name has stuck.
Anyway, the point is that all massless particles travel 300 million meters every second. The details of this are explained by special relativity, but simply put, it's physically impossible for a massless particle to NOT travel at 300 million meters per second. They can travel in a straight line or bounce off of things and change direction, but the speed of a massless particle never changes.
And so mass is just the property of NOT HAVING TO always travel at the speed of light. As a side effect, this also means not being ABLE to travel at the speed of light, but the key is that particles with mass are lucky enough that they get to travel at ANY speed they want – as long as it's slower than light. The amount of mass something has just tells how hard it is for it to change from one of these speeds to another. Now, in Part I we mentioned that if there were no Higgs field in the Standard Model, ALL particles should be massless and thus travel at the speed of light. But you and I and swiss cheese clearly have mass, because we have the beautiful luxury of being able to sit still.
So how does the Higgs field help us do that? Well, while massless particles can only travel at the speed of light, they ARE allowed to bounce off of things. Things like particles, which are really just excitations in a quantum field. For example, the electron field is more concentrated at certain places called "electrons" - and everywhere else is "empty space". But the Higgs field is unusual in that it has a high value EVERYWHERE – and to be clear, this high value is NOT the famous Higgs Boson - that's an extra excitation in addition to this already elevated field. But because the Higgs field has this everywhere non-zero value, any particle that CAN interact with it is pretty much bouncing off of it all the time. And if a massless particle bounces back and forth and back and forth (or, since it's quantum mechanics, does both at the same time), then even though in between bounces it travels at the speed of light, when you add everything up it LOOKS like the particle is going slower than light. Even... like it's not moving! And since only things with mass are allowed to not move, our massless particle now looks and acts like it has mass. Well done, Higgs!
What's more, the Higgs field can even interact with its own excitations, which is to say, it can give mass to the Higgs Boson, too. Actually, the Higgs field likes to interact with itself so much more than with the lowly electrons and protons that make us up, that the Higgs Boson has a great deal more mass – and that's part of why it's been so hard to find. But we shouldn't complain, because even though the Higgs has given us a lot of trouble and only a little bit of mass, at least we have mass, which allows us the simple pleasure of not moving."