、。
I was sleepy, so I went to bed.
Literal
Sleepy became [because-ので], I [topic-は] slept.
眠くなる = 'become sleepy' (adjective + なる). 寝る ('to go to bed, to sleep') is slightly distinct from 眠る ('to be asleep'): 寝る includes going to bed, while 眠る is the state of being asleep.