Could someone explain the significance of this, for those of us that don't know?
In particle physics there are a couple important fundamental principles, one of the most basic is the various conservation rules. Energy/mass, charge, lepton number, baryon number, etc, all of these things are conserved. But that leaves open a big window, because it means you can have any sort of particle reaction possible as long as you have enough energy and the various other "quantum numbers" (charge, leptons, whatever) are balanced. So, for example, if you have nothing more than high energy photons (gamma rays) you can create particle/anti-particle pairs easily (such as electrons/positrons) because in those situations everything that should be conserved is exactly balanced (since anti-particles have opposite charge, spin, lepton number, and so on). You can get more complex particle reactions so long as they are still balanced.
Now, let's say you want to study a particular particle. If those particles aren't naturally occurring, like protons or neutrons or electrons, then you'll need to figure out how to create them. And that means creating conditions where there's more energy available than the energy required for reactions that involve those particles. For example, the LHC was operating at 7 TeV total energy for collisions, which enabled them to discover the 125 GeV Higgs boson. Note that there's a significant difference in those energy levels (nearly a factor of 100), and this is because the process is somewhat inefficient and random and also there's some overhead in the reactions. For example, you need twice the energy of an electron particle in order to see electron-positron creation reactions, due to that balancing aspect.
Ultimately what you have is a huge collection of reactions due to particle collisions at a given energy level. And these reactions can be compared against the reactions you'd expect to see. With more energy in the collisions you push out the "search space" into different realms involving different particles and different phenomena. You can then use that data to perform "tests", by comparing what you actually see to what you'd expect to see given certain theories (such as the Higgs-mechanism theory), and in that way verify or falsify a theory. The more energy you have the easier it is to search for higher energy particles and phenomena.
The LHC energy boost will make it possible to explore new realms of physics. To either rule out the existence of potential particles at certain mass levels or to establish their existence, as with the Higgs-boson. It'll make it possible to narrow down some constraints on the Higgs-mechanism theory as well as potentially fill in some details (or rule out some possibilities) with regard to Dark Matter.
