When you slake something, such as a desire or a thirst, you satisfy it. A big glass of lemonade on a hot summer day will slake your thirst.
The word slake traces back to the Old English word slacian, meaning to “become less eager.” If you slake something, like thirst, you become less eager to drink. In other words, you are less thirsty. Like satisfy and quench, close relatives of slake in meaning, the word is used to indicate that a craving is made less intense by getting whatever it is that you crave.