Other answers are missing something — it’s not just that space is big and the stars are far away and the light gets diffused away. Imagine you had an infinitely big, infinitely old, unchanging universe. Like OP asked, in this situation any direction you look, you would end up looking at the surface of a star, some just very far away. (The stars being far away doesn’t reduce the brightness because there are a lot more far away stars than near stars in this picture, so the light adds up.) This obviously isn’t what we see, so at least one of the assumptions is wrong.
* If the universe isn’t infinitely big, there could be a “farthest” star, so you don’t get all that light added in from the very far stars.
* If the universe isn’t infinitely old, it could be that the light from the farthest stars hasn’t reached us yet, so there are still plenty of gaps between the stars we do see.
* If the universe is changing over time, not all stars are visible all the time because some expansion removes them from the visible universe.
The answer is some combination of 2 and 3, both of which are consistent with our current understanding of the Big Bang and the expansion of the universe. The point is that you need to have a picture in mind where we can only see a finite number of stars at any given time, rather than an infinite number of stars. Today we call this [Olbers’ Paradox](https://en.wikipedia.org/wiki/Olbers%27_paradox) but the question has floated around since the 1600s at least.
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