If you had empty space and then matter expanded from a point in it, then some of the matter would be in the center, seeing everything moving away from it. Some of it would be near the edge and see darkness filling half their world.
We see matter moving away, and it seems unlikely that we just happen to be so close to a center of the universe. So we look for other options that don't require us to be in such a special place. Since we already know that space and time can be curved, when we look for other options, there are lots of really interesting options.
If the universe were 1d you could imagine that at different times, the universe is a circle, later times it is a larger circle. So the radial direction is time, and space goes around in a circle. So what we expand into is ... the future.
If the universe were 2d you could imagine that at different times, the universe is the surface of a sphere, later times it is the surface of a larger sphere. So the radial direction is time, and space goes around in a spherical shell. So what we expand into is ... the future.
Since the universe is 3d we can imagine that at different times, the universe is the surface of a hypersphere, later times it is the surface of a larger hypersphere. So the radial direction is time, and space goes around in a hyperspherical shell. So what we expand into is ... the future.
The advantage of these models is that every point in the universe at a particular time is treated equally. We can then put in specific details about how quickly it expands, the matter and radiation and other fields in it and see what predictions it makes. If it agrees with what we see, we like it but keep checking. If it doesn't agree, then we try to fix it, look for errors, and consider alternatives.
Having the expansion be into the future is nice because that is what happens if you wait. Any part of the universe ends up in future, so maybe the future is just bigger than the present. But if you run that reasoning backwards, the universe used to be smaller in the past. Maybe it contracts towards a point, or maybe something interesting happened when it was small.
It's an area of active research.