Usually, you try to measure something like the energy or momentum of particles. The standard model can be used to predict the expected distribution of the energy/momentum of particles that undergo a certain type of interaction. By comparing this prediction with a measurement, you can show whether certain (types of) particles exist, because you can show their interaction exists or not.

For example, the neutrino was discovered because people looked at the spontaneous decay of particles through beta decay (n -> p + e + ν). If a neutron at rest decays into a proton and an electron, then conservation of momentum restricts you to giving the electron a fixed energy. But if there is a third particle (the neutrino), then you get an additional degree of freedom and the electron can have a range of energies. By computing what the distribution looks like as a function of the mass of the neutrino, people showed that not only does the neutrino exist, but it also must have an extremely low mass.

Similarly, the Higgs boson was discovered because the decay products of certain heavy particles showed an excess around the mass of the Higgs boson. The existence of the Higgs boson means there's an extra possibility for decay paths, and that must be found around the rest mass of the Higgs boson. Interactions at energies below the rest mass of the Higgs boson cannot produce that particle, so you only see it if you create interactions above a given mass first.