visualisation of the universe's expansion

You were correct, you teacher was incorrect.

It is the space that expands - much as a surface of a balloon does.

An explosion is a poor analogy in contrast because - as you suggest - it implies something to expand into.

For another way, not as accurate as the balloon analogy but maybe helpful - imagine being trapped inside an expanding loaf of bread in an oven. You have no knowledge of the world outside the loaf - and you may assume the loaf is infinite in extent. But as the dough rises the gaps inside the loaf get larger without the mass of the loaf itself increasing. You could compare our position in the universe to being inside such an infinite loaf.