This is the 17862nd most frequent Russian word.
уснуть
To fall asleep.
The word 'уснуть' is used here to describe the action of beginning to sleep or falling asleep.
Я не мог уснуть до полуночи.
I couldn't fall asleep until midnight.
The word 'уснули' (past tense, plural) indicates the process of beginning to sleep, referring to the children.
Дети быстро уснули после долгого дня.
The children quickly fell asleep after a long day.
The word 'уснёшь' (future tense, second person singular) forecasts the action of falling asleep in the near future.