There's no need to invoke something external for the universe to expand into. Our universe is probably finite but unbounded. Imagine 2D creatures exploring the surface of a sphere. Keeping on the same bearing, they will eventually return to their starting point; go where they may, there's no boundary.
We are 3D creatures, perhaps on the surface of a hypersphere, and would experience the same if we tried to cross the universe: there's no boundary for us to find, and a constant bearing would bring us home, but arriving from the opposite direction we started with.