According to the theory, the Universe is permeated by a field of Higgs bosons,...
Then I read that:
As the boson is unstable, it will quickly decay...
How can the Universe be permeated by a field of unstable, rapidly decaying particles?
They decay rapidly in the free state. When bound they live forever, nearly.
Good question.
Here we have to distinguish between real and virtual particles. Real Higgs bosons decay almost immediately, but the Higgs field is composed of virtual Higgs bosons. There are two reasons why virtual Higgs bosons don't decay in the sense that real ones do: 1) they don't exist long enough, and 2) there isn't enough energy available to create the decay products of a Higgs.
Virtual particles are hard to visualize. On the one hand, they aren't "really there", in the sense that there isn't enough energy available for them to exist, and they can't be manipulated like a real particle. On the other hand, they "really exist", in the sense that they do exhibit a subtle--or even a strong--influence on whatever physics is taking place.
The canonical visualization centers on the Heisenberg Uncertainty Principle. It states that the uncertainty in energy times the uncertainty in time is intrinsically greater than some tiny quantity. One implication is that you can "borrow" an arbitrary amount of energy from the vacuum, provided you "pay it back" in a brief enough time that the inequality is satisfied. The more energy you borrow, the faster you have to pay it back, but as long as the HUP is respected, you won't violate any conservation laws. Since, in physics, whatever is possible is compulsory, the vacuum is therefore a boiling sea of every possible type of particle popping briefly in and out of existence.