Space does get bigger, but that's separate. What eternal inflation is claiming is that some fourteen billion years ago there was a false vacuum collapse, and we're living inside the bubble that resulted, but said bubble is itself still expanding.

It won't ever get anywhere, since space outside the bubble is inflating much, much faster than inside. It means there's a border to our universe, which is expending at lightspeed, but it's far outside the Hubble horizon and thus entirely unreachable.