Cosmic rays are completely ionized atomic nuclei (without any orbiting electrons) that travel through space at very high speed. We already talked about how they are distributed in LEO orbit (as Space Station and Shuttle orbits) where their intensity increases near the magnetic poles and that there is a trapped component of protons in the inner radiation belt that is crossed by LEO orbits in the South Atlantic Anomaly region.
But what happens when cosmic rays go through matter? Ordinary matter is composed by the same nuclei of the cosmic rays, but in this case the electromagnetic strength binds them to their electrons in order to form atoms. The total charge of electrons is equal and opposite to the charge of the corresponding nucleus in order for the atom to be neutral. The nonintutive fact is that ordinary matter is basically empty. If we look at the simplest atom possible, the hydrogen one, we found a single electron orbiting a single proton (in the classic description): the atom dimension is given by the radius of the electron orbit that is 5.3 * 10-11m, while the nucleus, a proton in this case, has a radius of 10-15m.
To understand the relative dimensions, if you imagine the proton as big as a golf ball (that is about 4 cm) the electron would orbit 1 km far away. The proton mass is 200 times the mass of the electron, so for a 1 kg proton the corresponding electron would weight half a gram. Heavier atoms have slightly different ratios, but the orders of magnitudes are similar.
After this check we are entitled to say that matter is empty and mass is enclosed in a very small space (all in nuclei). So whenever a charged particle (from cosmic rays) crosses matter it has much more probabilities to interact with electron distributions than with nuclei themselves. The particle will lose part of its energy during this interaction and its trajectory will be deflected. These two effects are mainly due to
- anelastic collisions with atomic electrons
- elastic collisions with nuclei
Less effective processes are
- Cherenkov radiation emission
- nuclear reactions
- bremsstrahlung
These interactions happens multiple times during the travelling path and their effect cumulate and cause a total energy loss and deflection. The energy loss by impinging particles is transferred to atoms causing their excitation or ionization. The amount of energy transferred in each collision is a very small fraction of the total kinetic energy of the impinging particle. But the number of collisions is so high that even a small thickness causes a considerable energy loss. A 10 MeV proton loses all its energy in 0.25 mm of copper. The high number of interactions in a macroscopic thickness ends up in reduced statistical fluctuations and it permits to define a mean energy loss for lenght unit, that is called stopping power or dE/dx. In the next post I will talk more about mean energy loss, and we will discover which are more dangerous between low or high energy cosmic rays and why it is so difficult to shield from them.
ALTEA Space 
Well I disagree that most of matter is empty space. The space around a nucleus is filled with a field. What might be mass at one temperature is not mass at another. In other words, matter is a bit more than mass alone.
Of course this is an oversimplification. There is no perfect empty space or vacuum, but anyway it is a good picture. The electron wave function amplitude is maximum near the classical orbit and almost zero near the nucleus.