The rules in these few chapters are archaic at first (and even second) glance: some still make sense, though, if you take a step back and try to look at the underlying reasoning.
For example: what do you do if you find a body in a field?
In the myriad of modern-day crime novels, movies and TV shows, you would put your best detective on it, dither around for a while as a few more people were murdered, then solve the case by good luck or sheer genius. The murderer, who by now has become a serial killer, would end up in gaol. Even cold cases (old, unsolved crimes) are solved this way. It’s a testament to the cleverness and dedication of modern policing, if you ignore the existence of cold cases in the first place.
Without the benefits of a friendly scriptwriter and modern forensic technology, not all crimes in ancient Israel were so neatly solved. Furthermore, murder was not just against the Law and a crime against a person, but also a crime aganst humanity, the land itself, and against God.
For this reason, unsolved murders need a way to take away the moral and spiritual guilt. The elders and judges were to deal with it by going to the nearest town and having the elders (of that town) select:
Deuteronomy 21:3–4 a heifer that has never been trained or yoked to a plow. … and break the heifer’s neck.
The priests, who were the nation’s judicial system, were then to pronounce a blessing, and the elders were to wash their hand over the heifer and say:
Deuteronomy 21:7–8Our hands did not shed this person’s blood, nor did we see it happen. O Lord, forgive your people Israel whom you have redeemed. Do not charge your people with the guilt of murdering an innocent person.’
The community would be absolved from crime of the murder and also cleansed from the spiritual guilt of murder.