The universe is everything. There's no outside the universe, since whatever that "outside" was would also be part of our universe. It's not empty space, since that too would be part of the universe. There is no edge of the universe since that would imply that there is something beyond that edge. In scientific terms, the universe is homogeneous and isotropic.

We've proved the universe itself is getting bigger, which means that yesterday it was smaller and the realization of that fact was how the big bang theory came to be.

There are lots of resources out there if you want to build an intuition around how a universe that's everything can get larger without needing something to expand into.

You can imagine the balloon’s curved surface geometry without assuming that it is embedded into a 3D space. This is what Riemann discovered in the 19th century. Such a non-embedded space is called a manifold. In the case of the balloon, it’s simply a non-Euclidian two-dimensional space. A third dimension doesn’t enter into the picture in that formulation.

Four-dimensional spacetime is considered such a manifold. Its three-dimensional space “slices” are predicted to expand with time by general relativity (first predicted by Alexander Friedmann in 1922), and actual observations have confirmed those predictions.