There's no notion of spacetime outside the universe. The spacetime is the universe.

Two assumptions:

a) the universe is infinite. In this case it doesn't expand into anything, the things inside of it just stretch apart.

b) the universe is finite. In this case it expands into something, but because we're trapped inside spacetime, we can't answer this scientifically.

The wording "expanding" adds more confusion than it helps. See [1] for the discussion about this topic.

[1]: http://curious.astro.cornell.edu/ask-a-question/104-the-univ...

I think it’s like the universe being a balloon (infinite or not) and inflating it. The space between any two points on the balloon increases over time. And at the Big Bang the space was basically 0.

Think of it another way: All matter in the universe is getting smaller. But the universe have the same size, so distances between everything increase.

the universe doesn’t expand into something bigger. It’s like multiplying the set of the real numbers by two, the set doesn’t become bigger and its boundaries do not change.