disclaimer: I'm not a particle physicist so I might get some of the details about the CMS experiment or particle physics wrong but the detector physics should be ok.
The previous answers have answered well the question of how the footprint of the final detected particles can be used to infer the properties of the initial particles of interest. I'll try to give a bit more information about the process of converting the presence of a particle into a detectable signal which can actually be plotted on a computer screen.
I think one of the most basic examples of particle detection is single photon detection with a photodiode. Take a single photon avalanche diode for example. Abstractly, a single photon hits the semiconductor material and is absorbed to create an excited electron. The electron then travels through the semiconductor (driven by an electric field produced by the diode bias voltage) knocking free other electrons on its way causing a cascading current which gets larger and larger. The experimenter then runs this current through a resistor (transimpedance amplifier) to create a voltage and then measures this voltage with some sort of oscilloscope or voltmeter. Whenever the experimenter sees a spike in the voltage she can infer the presence of a photon at the location of the detector.
Here's a little more information on the quantum mechanics of photodetection.
There are a wide range of different types of detectors but the fundamental idea is at the core they utilize some process where the detectable particle* is converted into an electron or a burst of electrons which are then amplified into a detectable current or voltage and then use this voltage to infer the presence of (and sometimes the energy of) the particle of interest. In addition, by placing many such detectors in some spatial pattern you can learn even more information about the detectable particle (such as its trajectory or momentum) by looking at the spatial pattern of detectors which are 'lit up' by the detected particle.
So basically for any 'event' (particle shower from protons colliding) the experimenters raw data is a time trace of the voltage coming from each one of the millions** of single particle detectors that make up the CMS detector. Then in an extremely computationally taxing analysis process all of these signal channels are analyzed and put together into a picture that the experimenters can make sense of and further analyzed to determine what sort of interaction created the particle shower which was detected. In the case of CMS this all takes the work of thousands of scientists and engineers.
*A quick scan of the CMS wiki tells me there are detectors for electrons, muons, photons hadrons (protons, neutrons, kaons, pions) for example.
**Woah I just learned how many detectors or 'pixels' make up the detector, that's a lot!