Muons are produced from the interactions between an energetic proton beam and a light-element target. The proton collisions produce pions, which decay into muons. The muon beam produced can be 100% spin polarised (the polarisation direction is anti-parallel to the muon's momentum direction, as shown in the figure).
The muons are implanted into the material under investigation, where they come to rest. Local magnetic fields, for example from near-by magnetic atoms, cause each muon to precess.
After an average lifetime of 2.2µs, each muon decays and emits a positron. The decay positrons are preferentially emitted in the muon spin direction - they are detected, and tell us about how the muons' spins were behaving in the sample. This in turn tells us about the muons' environment - so we can learn about the atomic level magnetism, for instance, in the sample.

Shown here is a summary of the life of a muon used for exploring materials! It's fast and exciting!