Well you have two effects.
For starters, free neutrons are unstable, and they decay to a proton and an electron, with a half life of about 11 minutes. So then you have two charged particles that fly off in opposite directions wreaking havoc.
Secondly, neutrons, being uncharged, are not deflected by Coulomb forces, so they can score a direct hit on the atomic nucleus, and be captured. This then usually results in a proton being ejected from the nucleus, so the atom changes to a different element, of one less atomic number.
The ejected proton ("knock on" proton) also has high kinetic energy; often around 14 MeV, so it is the energetic charged particle products, that do the damage.
Neutron damage in biological tissue, is a strong function of neutron KE. Special "tissue equivalent" monitor detectors, are used to monitor neutron hazards, in locations where they could be produced. Impacting Deuterons, on a heavy ice target, will result in neutrons, or protons, in the 14 MeV range. The detectors, often proportional gas counters, containing organic materials either solid or gas, or both, that collectively mimic the capture cross-section for neutrons in biological tissues. These detectors, are like Geiger counters, but below the critical avalanche Voltage range