RobH_en_tn/deu/19/05.md

For example, if a man goes into the forest ... to cut wood

The author gives a hypothetical situation where a person accidentally kills another person. (See: rc://en/ta/man/translate/figs-hypo)

the ax head slips off of the handle

The iron part of the ax comes off the wooden handle.

strikes his neighbor and kills him

This means the head of the ax hits and kills the neighbor.

to one of these cities and save his life

It is implied that the family of the dead man may try to get revenge. The man who killed him can flee to one of these cities, and the people there would protect him. (See: rc://en/ta/man/translate/figs-explicit)