Higgs particle is one of many particles in the "Standard Model" of particles that make our universe. The importance of the Higgs particle is that, it "gives" mass to almost all the other particles. Without the Higgs particle, anything in the universe will be massless, and the universe would be very different. But I would not call the Higgs particle the "fuse" of the big bang, because the big bang happens at a much, much higher energy scale than the Higgs particle, and physics at that energy scale is not understood yet. The Higgs particle starts playing its role AFTER the big bang has happened.
Not being a theoretical physicist I can't explain it in its entirety, and even a physicist would need to use fairly sophisticated mathematics to describe quantum field theory that are beyond anything you'd see in 12th grade. However, I will give the incomplete and possibly out-of-date description that I know:
The standard model of quantum mechanics, which at present attempts to explain all forces except for gravity, predicts the existence of a Higgs field that is responsible for particles' inertia (i.e. the extent to which they resist force when being accelerated). This field, like all physical fields, would be represented by a carrier boson, just like the fields of the electromagnetic force are conveyed by photons, the weak force by W and Z bosons, and the strong force by gluons. This boson carrying the Higgs field is the Higgs boson, and the discovery of this boson in high energy physics experiments confirms the existence of the Higgs field, shedding some light on why inertia and mass are the same.
I don't understand why it's called the 'God particle'; I suspect that this is a misleading media description of it. Inertia is important in three of the four fundamental forces (electromagnetism, weak, strong), but the gravitational force does not care about inertia, since it affects the shape of space and time itself instead. However, as I understand it, it only impacts inertia, not the forces themselves, so to call it the 'God particle' is misleading.
The standard model does not deal with gravity at all, and the big bang and subsequent expansion of the universe is a gravitational phenomenon. As such, based on the physics that we understand, the Higgs boson would have nothing to do with the big bang. It is likely that gravity does have a quantum nature, but at the moment we have no means of testing any theories of quantum gravity, so there is no way to know if they are correct.
Finally, while the standard model did successfully predict the existence of the Higgs boson, it has other problems. For example, the standard model predicts that neutrinos should have no mass - but it turns out that neutrinos do have mass. Once again, these theories are incomplete, but so far as we know, the Higgs boson has nothing to do with the big bang.
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