Why did contact with a dead body make a person unclean?
You should remember that the kind of defilement brought by a human corpse was different from that brought by that of an animal. If you touched a dead animal, you must wash with water and remain unclean until evening.
But touching a dead human body resulted in seven days of uncleanness with a special ritual (involving the ashes of a red cow) to be performed that was not performed for any other uncleanness.
Death was never the intention of God for mankind. Death is the penalty for sin. As such, when death occurs in the camp, it is a reminder of sin. Cleansing from sin can only be accomplished by the offering of blood. The red cow was burned in its entirety (blood and all – the only time that happens) outside the camp. the one who burned it, the one who gathered the ashes, anyone who came into contact with it was unclean because of their contact with an animal. But the ashes of that red cow, offered for sin (vs. 9), reconstituted with water, cleansed those on whom it was sprinkled.
The ceremony was a reminder to all of Israel that death was bad, a consequence of sin. It may not be your personal sin that caused the death, but the existence of death itself was because of sin and death defiled anyone who came into contact with it.