I've been using spaces for a while, but I am genuinely still baffled by it. Intuitively, a space should be where I group rooms or people in a non-overlapping way, and cleanup the side-panel clutter... But I can access all rooms from all spaces, so what is it's purpose exactly other than a namespace in which to connect to other places?

The most important thing in my view is that you can invite other people to the space, and you can use that membership to control access to multiple rooms.

So let's say I were using Matrix for the hackerspace. With spaces, I can invite a new member to the space and they can join whatever associated rooms they want on their own.

Without spaces, I'd be stuck either inviting people individually to every room, or making every room completely public, neither of which are particularly palatable.