Ionization cooling is a technique for reducing the emittance of a muon beam. It is the only known technique for doing so that operates fast enough so the muons don't all decay away (some do, and the challenge is to minimize it).
This is called "cooling" in analogy to the thermodynamic cooling of a refrigerator, which reduces the phase space volume occupied by the molecules of a material. In neither case is Liouville's theorem violated by the system as a whole, as some other substance is heated (for a refrigerator that is the room air; for a beam it is the electrons in the absorbers).
The basic idea is to put the beam through a series of cells, each containing an absorber and an RF cavity, such that the energy the beam loses in each absorber is restored by the RF. As each beam particle traverses the absorber, it loses energy, so that both its longitudinal and transverse momenta are reduced. The RF cavity only replenishes the longitudinal momentum, so the transverse momentum is reduced.
In practice, many cells in series are needed to achieve the overall emittance reduction required for a muon collider. Note also that only transverse angles are cooled, so in practice the system has focusing such that transverse size is also cooled; the system is then arranged along a helix, which provides emittance exchange so that longitudinal emittance can also be cooled.
The energy loss contributes to the cooling of the beam, but multiple scattering contributes heating. The ratio between them is material dependent, and the best material is hydrogen, with higher-Z materials decreasing rapidly in cooling effectiveness.
The original document is available at http://www.muonsinternal.com/muons3/Ionization+Cooling