That's not exactly what the p-value or the number of sigma means; it's more that if the Higgs really did not exist, a bump of the observed size would only have such-and-such a chance of appearing at random. To translate that into a degree of certainty that the thing exists, you'd need some kind of prior probability estimate.
Now, in this case, most particle physicists were in fact pretty sure that something like the Higgs existed, and fans of supersymmetry additionally predicted that it had something like the mass observed. So the prior wouldn't, I suppose, be all that low, except for the mavericks who were predicting it didn't exist.
But this particular conceptual slippage can be crucially misleading when one is talking about tests for things that are very unlikely in the first place (e.g. lab tests for rare diseases). If the p-value is one in a thousand but the thing was one-in-a-million unlikely to begin with, the evidence doesn't mean you've got something with 99.9% confidence.
Also, I get mildly peeved by articles saying that the Higgs is responsible for all the mass in the universe. The Higgs field gives mass to quarks and leptons. But actually more than 99% of your mass, in your protons and neutrons, comes from a completely different field that has to do with the strong nuclear force. The proton or neutron mass is much larger than the combined masses that the individual quarks get from the Higgs.