Clean experimental conditions are to be understood with respect to other types of colliders.
Example 1: An event with 2 muons in the CMS detector at the LHC (in a configuration with still moderate background conditions). Each beam crossing comes with 30 simultaneous gluon-gluon interactions, out of which one may (or may not, more often than not) be interesting. The interesting gluon gluon interaction also comes with its "underlying event" made of the debris of the colliding protons.
Example 2: An event with two jets in a linear e+e- collider. There is much less pileup events than in proton proton collisions, and no underlying event. However, the beamstrahlung arising from each beam crossing consists of many photons and e+e- pairs, leading to important backgound in the detector. This background places specific technological requirements on the detector to be identified offline.
Example 3: Four different Z decays in a circular e+e- collider (here, LEP): no pileup interactions, no beamstrahlung, no underlying events. The events are clean and simple, and the requirements put on the detectors are much less. Another important advantage over proton proton collisions (related to the absence of underlying event and of beamstrahlung) is that the total energy and the total momentum in the final state is the same as in the initial state, i.e., E = 2 Ebeam and p = (0,0,0). These conservation laws allow precious kinematic constraints to be used when it comes to measure resonances masses, missing energy, and the like.
In conclusion, what collider do you think brings the best (in particular, cleanest) experimental conditions ?