If the universe is expanding, what is it expanding into?
It is expanding into nothing.
Technically physicists all over the world have been working to solve the mystery of origin of our Universe.Various models of theory of Big Bang have been developed over time.
It is considered that Universe at its beginning was an infinitely small ball(of zero radius) with an infinite density. As it began to heat up, it began to expand. Though scientists like Albert Einstein didn't agree that the universe is expanding, but several years later, several probes actually confirmed that it was and that too at an accelerating rate.
But since Big Bang was considered to be the beginning of everything, it was believed that there's nothing beyond it and that the idea of thinking about what was before big bang seems unnecessary.
Therefore, though there might be something beyond our universe, theoretically it is considered to be expanding into nothing.
The Universe can be thought to be expanding into a state of maximum entropy
The rate of expansion of the Universe is believed to be increasing based on the empirical studies of the supernovas. This means that the Universe will continue to expand infinitely. As the universe expands matter and energy will become increasingly spread out and diffused. Eventually matter will become so spread out that there will be no matter only energy. Energy will move to the lowest possible level where there is no usable energy left. The universe will have effectively ended.