Neutrinos detection. What mechanism for their interaction with matter?

There are two ways to detect a neutrino:

1. The neutrino "collides" (i.e. interacts via the weak interaction) with a charged particle. In the interaction some of the neutrinos momentum is transferred to the charged particle. if the neutrino is sufficiently energetic and the charged particle is sufficiently light (e.g. an electron) then the charged particle can be accelerated to relativistic speeds and will emit Cherenkov radiation as it interacts with other particles in the detector medium. This Cherenkov radiation can be detected.
2. The neutrino interacts with a heavier neutral particle, such as a neutron. The neutron is transformed into a proton, and to conserve charge an energetic charged particle such as an electron, a muon or a tau (depending on the flavour of the neutrino) is created. Again, the electromagnetic radiation given off by this energetic lepton can be detected.

Both of these detection methods depend on weak interactions which are very rare, so a high neutrino flux and a large detector volume is required to produce a useful number of interactions in the detector. Fortunately the flux of solar neutrinos is sufficient to produce useful detection rates. Detection of neutrinos that have originated outside of the solar system is still a very rare event.

The Wikipedia articles on neutrino detectors and neutrino astronomy provide more details.