It's essentially statistical. First, the beam. The beam at a collider is actually made out of longitudinal 'buckets' of particles. The buckets are timed to cross at the interaction point inside the detector, where magnetic lens arrangements squeeze the beam as far down in physical size as is practical. By controlling the beam, you know when exactly to look for collision products, and expected collision rates are calculated from particle distributions and cross-sections.
Next, the detectors. There are a variety of detectors around the interaction point that serve different purposes. Broadly speaking, they measure deposited energy, and in which direction it is going. That's what physicists are interested in, so they can work out what those particles were.
Everything at an experiment like this is modeled and calculated. At the very bottom you have fundamental physics, particles being created and destroyed, modeled in a simulation tool such as Pythia. Once you know what's supposed to come out, you can simulate how the products behave in your detectors, by building a model in a tool like Geant. If you model from end to end, you essentially test physics theory against the hard bedrock of reality.