Neutrino mass is less important than their lack of a charge. Most matter interactions, after all, are electromagnetic repulsion between electrons in atomic orbits.
Since atoms are mostly just empty space and neutrinos have no charge to attract or repel them from either the electron shell or the protons in the nucleus, they just pass right on through.
The low mass of a neutrino does mean that it can move faster. Accelerating a non-zero mass toward the speed of light requires more and more energy scaling to infinity, so you can never quite get there, but the low mass of a neutrino lets it get pretty close.
Photons, of course, are absorbed by atoms and re-emitted (in transparent substances anyway), and this slows light down (even though the actual photons always move at
c, every time they strike another atom there's a delay as it's absorbed and re-emitted). It's why
c is specifically the speed of light
in a vacuum.