A Deathstill was a place where the water of a person's dead body could be reclaimed. On Arrakis, water was the most precious commodity. According to Fremen tradition, a man's water belonged to his tribe. As a result, the Fremen developed deathstills as a way of reclaiming the water of their dead, since it belonged to the tribe, and the dead had no further use of it.
The Fremen also put the deathstills to use for reclaiming the water of their enemies. It was usual practice for Fremen women and children to go through a battlefield when fighting was over, killing the wounded soldiers of the enemies with their crysknives, so that their water could be reclaimed in the deathstills.
Presumably, the water reclaimed from the deathstills would be used to water the sietch, and possibly go toward the terraforming of Arrakis begun by Imperial Planetologist Pardot Kynes. It was stored in catchbasins.
The only way in which someone's water would not be reclaimed by the tribe through the deathstills is they were found guilty of something so heinous that their water was considered tainted. An example of this was the people of Sietch Jacurutu, and Alia Atreides (whose water was poured onto the sand due to her possession).