Welcome to the forum.
It's not really possible to give a "layman's terms" answer because the situation involves technical aspects. It's like asking to explain parking a car without referring to the motion of a car.
But here goes.
The basic idea of particle colliders is to adjust the energy and see what comes out. At a really simple level, as you slowly increase the energy you get more kinds of interactions.
Think about colliding two protons just for example. They will have an energy available due to their kinetic energy. The mass of an electron is a certain amount. When the energy available in colliding two electrons becomes greater than twice this mass (refer to relativity and equivalence of mass and energy) then the collision can spit out an electron and a positron. And you can see these fly away. When that happens you see the chance of a reaction increase strongly. This chance is expressed as a cross section (an area) and the cross section is a function of energy. When you reach twice the mass of an electron you see a sharp increase in cross section. This is usually called a resonance.
The same happens when you get enough energy to create a proton-anti-proton pair. Or a neutron-anti-neutron pair.
Now the complicated bits. When you get enough energy to produce a quark-anti-quark pair, you can make such a pair. And you see another resonance. And this pair comes out as an unstable particle. You can infer the characteristics of this particle by what it decays into, and how long that takes. You scan energy, see the increased reaction cross section, collect the decay products, and infer what decayed. And you scan up in energy and infer that there are six quarks with particular symmetries.
Now it's a very complicated business because there are quite a few combinations of quarks, in a variety of energy states, that can be produced. And they can decay into a variety of end products. And the inferring of characteristics from those decay products is pretty complicated.
So, fundamentally, it's banging things together to see what parts come out. And inferring the structure from the characteristics of those parts.