After 400 years of debate on the question, there is now fairly wide agreement among astronomers: there just aren't enough stars in the observable universe to fill up the night sky.
We don't really know how many stars there are. What we do know is that however many there are, we can see only a finite number of them.
The oldest stars are about 10 billion years old, meaning that the greatest distance starlight can have traveled is 10 billion light-years. So the only stars we could possibly see are those within a 10 billion light-year radius of us--the light from stars farther away has yet to reach us. The few jillion stars in our corner of the cosmos (AKA the "observable universe") don't have the collective candlepower to illuminate the night sky. (True, as time goes on, light from more distant stars does reach us, but meanwhile some close-in stars are dying out.)
So that's why the night sky is dark.