This is a great explanation. One other bit I'll add is that the particles they're colliding are protons, which are made of other particles, called quarks. While the protons are colliding with a relative energy of 13TeV (or will be) the actual collision energy depends on the collision angle of the quarks inside the proton. If it's a glancing collision, which it almost always is, the actual collision energy is much lower. In other words, we don't get to make particles w/ masses of 13TeV. The Higgs for reference has a mass of 125 GeV. That's one reason physicist are considering using Muons in future accelerators instead of protons. They're not made of other stuff.
One of my favorite videos on this topic was a Great Courses video by Sean Carroll. Well worth the $40. http://www.thegreatcourses.com/courses/the-higgs-boson-and-b...