It doesn't. The Higgs mechanism is said to do be responsible for mass, but take a look at A Zeptospace Odyssey: A Journey into the Physics of the LHC. It’s by Gian Francesco Giudice, a CERN physicist. There's a search-inside on Amazon. If you search on Higgs sector you can read pages 173 through 175. He starts by saying: “The most inappropriate name ever given to the Higgs boson is 'The God particle'. The name gives the impression that the Higgs boson is the central particle of the Standard Model, governing its structure. But this is very far from the truth.” He also says the Higgs mechanism is “the toilet” of the standard model, and is “frightfully ad-hoc”. On page 174 he says: “It is sometimes said that the discovery of the Higgs boson will explain the mystery of the origin of mass. This statement requires a good deal of qualification.” He ends up saying “In summary, the Higgs mechanism accounts for about 1 per cent of the mass of ordinary matter, and for only 0.2 per cent of the mass of the universe. This is not nearly enough to justify the claim of explaining the origin of mass.”An 'explanation' of mass said that there are Higgs Bosons everywhere, creating the Higgs field that creates mass. However, the Higgs Boson has an incredibly short life span, before it 'transmutates' into other particles.
How does that work?
The CERN press office don't correct the garbage you read because it suits them if you think this is something more important than it is. E=mc² is responsible for mass. If you heat a container of gas, you increase its mass. If you trap a massless photon in a box, you increase the mass of the system.
