As you probably know, in high-energy hadron collisions, you typically have a quark in one of the incoming hadrons scatter off a quark in the other incoming hadron (deep inelastic scattering). For the brief moment in which the quarks are interacting, they can be considered as free particles to some extent, so they "bounce" off each other and emerge back-to-back along a random axis. Since this typically knocks the quarks out of their original particles, you'll get two jets of daughter hadrons, one for each of the scattered quarks.
Jet quenching can occur when this process takes place in the presence of a "soup" of many other hadrons. Typically, what happens is that the quark scattering takes place near the edge of the hadron soup. One of the jets will be directed outward, towards lower particle densities, and the other will be directed inward, toward higher particle densities. The latter jet will lose most of its energy due to further collisions and gluon radiation (it gets "quenched"), but the former jet continues out to the detector. So what you see is only one jet of daughter hadrons.
Several years ago I wrote a research paper about calculating the drag force on a quark in a quenched jet. Figure 5 on page 12 shows jet quenching in action. (I'll include the figure itself here once I get back to my other computer which has the source graphic on it.)
The analogy between jet quenching and hydrodynamics is not exactly in my area of expertise, but from what I've heard, hydrodynamic models (developed for classical liquids) have had some success describing certain properties of the quark-gluon plasma. It has been said that this suggests that the QGP is actually more like a liquid than a plasma, but I'm not sure to what extent that's literally true. You don't need the hydrodynamic analogy to explain how jet quenching is able to occur, though, at least not at a qualitative level.