"Kill the crew of 20 ships by sector 6 in the Mantis Cruiser."
Unlocking this achievement requires the entire enemy crew of each ship to be killed either by boarding personnel, burning, suffocation, or some other method that leaves the enemy ship intact. Simply destroying an enemy ship with its crew inside does not count.
"By sector 6" means the crews have to be killed by the end of the sixth sector, not by the beginning of it.