The universe doesn't have to expand into anything, it's a feature of non-Euclidean geometry and general relativity.
Even an infinite universe can expand in a sense by moving everything apart from everything else.
When studying general relativity, the geometry is intrinsic to the space or space-time being studied, and doesn't need to be imagined in a larger space.
Welcome to the June 2023 Ask Me Anything episode of Mindscape! These monthly excursions are funded by Patreon supporters (who are also the ones asking the questions). We take questions asked by Patreons, whittle them down to a more manageable number -- based primarily on whether I have anything interesting to say about them, not whether the questions themselves are good -- and sometimes group them together if they are about a similar topic. Enjoy!