Unbound neutrons can collide with nuclei in a target materials and result in a chain reaction of potentially harmful effects.
The neutron will impart a large amount of kinetic energy to a nucleus it collides with causing it recoil and become ionized. This heavy charged ion (HCI) can then collide with other atoms creating a branching cascade of collisions that continues until all the energy is dissipated. So for every nucleus a neutron collides with, you get a shower of HCIs that are very effective at break molecular bonds, either damaging DNA directly or by creating free radicals that later attack DNA. Depending on its energy and the mass of the initial recoil nucleus, a neutron may still have enough kinetic energy to create additional cascades.
If a neutron is absorb by a nucleus, that nucleus may be transmuted into a radioactive isotope that may decay by alpha, beta, or gamma emission.
Neutrons are extremely penetrating as they are not subject to the electric force and will only react with nuclei (much smaller than atoms) and most nuclei have very small neutron collision cross-sections. Hydrogen doesn't have a big cross-section, but it has a high number density to weight ratio for water and hydrocarbons, so neutrons are more likely to dump their energy in them. Lead in contrast has a tiny cross section and few nuclei per unit mass and is almost transparent to neutrons.
Points 2 and 3 are actually why the neutron bomb was developed. The radii at which the blast, heat, or radiation from a tactical nuclear weapon are sufficient to knock out a tank, armored vehicle, bunker are actually not that large. A neutron bomb is a nuclear weapon optimized to maximize neutron flux, which goes right through the metal and composites used to make a tanks armor doses its water laden crew. Some of the nuclei in the vehicle will also become activated and add to the dose. The crew will succumb to acute radiation poisoning and the vehicle will become a hazard until the activated nuclei decay.