The invariant mass is the rest mass of the particle (i.e. it's "inherent" mass). You can calculate it by taking the final state decay products of the original particle (i.e. the particles that are actually observed by the detector) and summing up their four-vectors (squared).
You can plot the invariant mass calculated from any particular final state, and for a rare particle like the Higgs the majority of the contributions to your plot will be from background processes (i.e. not Higgs decays) that decay into the same final state.
If you have a lot of Higgs decays in your sample you should be able to see a clear peak in the distribution at the invariant mass of the Higgs boson, a clear sign that the Higgs (or something with the same mass) exists.
Often by the time the discovery has reached statistical significance, you might not really be able to see such a clear sign in the mass distribution. I.e. the calculations are telling you it's there but you can't see it that clearly.
I wouldn't really say this helps confirm the discovery in a scientific sense, just that it's reassuring that the signal is so strong that you can see it by eye.