The clearest framework to study and interaction is the center of mass system. To go to the center of mass one has a Lorentz transformation. All vector components in the direction of the velocity vector entering the transformation are changed by the transformation. The components perpendicular to the transformation vector are not affected.
Thus p transverse is the same either at the center of mass or in the laboratory system. So it is a snapshot at the time of running the experiment, of what is going on at the center of mass without having to go to the details of the event. The same is true as a first filter in the subsequent analysis for picking up subsets of data that have events of interest.
In experiments one searches for heavier and heavier mass resonances of particles in the interaction, searching for new physics. These will be decaying into high momentum fragments and thus a high p transverse cut picks at the raw data level events that have a probability of coming from the decay of a supersymmetric particle, for example.