Here is one way to comprehend this. The universe is not expanding "into something". Space itself is expanding, but this is not so easy to understand for us, since we are 4 dimensional creatures.
However, we can make a stab at it using a crazy but interesting analogy. There are 4 dimensions that we can experience: for anything to exist for us, it must have height, width, thickness, and it must exist for some length of time (that is why time is called a 'dimension'). But science indicates that they are likely 5 (or many more) dimensions. For us to understand any additional dimensions, we must stretch our brains and imaginations to the max.
So imagine a new "fairy tale" world where all the beings were only 3 dimensional: they had height, width, and time, but no thickness. These beings were thinner than a piece of paper and they lived on a flat plane (not an airplane of course, but a very large flat surface like a giant piece of paper). The beings lived on the plane and were able to move about, but only in two dimensions: forward-backward and left-right, but they could never leave the plane or comprehend that there may be another dimension (thickness) above or below their planar world.
Now imagine that the plane upon which the flat beings were living on was not really a plane, but was actually a very large sphere. To us, we could see that it was a sphere, but the flat beings were small and the sphere was very large so they still thought they were living on a very large flat plane. To them, there were some very odd things about their world. For example, if one of them were to start in a particular place and move exactly straight for a long, long time, the being would end up in exactly the same place it began. This was odd to them, but still every measurement they could take seemed to indicate that they lived on a large flat plane, not a sphere (plus, a sphere is a 4 dimensional concept and they could not comprehend additional dimensions very well).
Now imagine that the very large sphere was like a balloon that was being filled with air. The sphere was stretching and expanding everywhere: the very material of the sphere was stretching and the 'space' of the planar beings was expanding at all points simultaneously. Note that their planar universe was not expanding "into something". Their planar space itself was expanding. To us, as 4 dimensional beings, this was easy to see and comprehend from a perspective above their world, but the 3 dimensional planar beings could not figure out where this additional planar space "came from." Their entire and very existence was confined to their planar surface--they could not comprehend any other 'space' outside of it.
So, to end the analogy, we are very similar to those flat beings living in their 3 dimensional world. Our very space (4 dimensional space) is expanding all around us, but this is hard to see or understand clearly unless one can exist and observe it from a 5 dimensional world (time, height, width, thickness, and another dimension{s}). But just as we can understand and oversee the 3 dimensional planar world expand, a 5 dimensional being could easily understand our universe expanding in the same way.
Hope that helps slightly. Any helpful explanation of this concept will be a mind bender.
Where would you want to swim ?