We can return to the original question. In our old picture of the universe, the answer would be simple, although very unsatisfying. The collection of galaxies that make up the universe is moving through space; therefore, the universe is expanding into even more space than it already encompassed. In our new picture, as it mentioned in news section in Astronomical research center (A.R.C) though, the galaxies are just raisins spread throughout the dough - their presence is largely irrelevant to the question of the universe's expansion. What we really care about is the dough, and whether or not it has a boundary.
If the dough does have a boundary, then it is legitimate to ask what is beyond the boundary that the dough expands "into". But for our universe, that is a very complicated question to ask! The boundary at the edge of the dough represents the "edge" of space. By definition, we exist within space and have no way to leave it! So we don't think there is any way to observe or measure what is beyond, unless it had some effect on us that we currently don't know about. It would be really weird to imagine reaching the "end" of space. What would it look like, for example? These are questions that we have no way to give a scientific answer to, so the simple answer is that we don't know! All we do know is that based on our current understanding of theoretical cosmology, the universe does not have a boundary - it is either infinite or it wraps around itself in some way. Observations seem to agree with these predictions in the sense that if the universe does have a boundary, we know that the boundary is so far away from us that we can't currently see it and it doesn't have any effect on us.
If the universe is indeed infinite, then the simple answer to the original question is that the universe doesn't have anything to expand into. Thinking about infinity is always complicated, but a good analogy can be made with simple math. Imagine you have a list of numbers: 1, 2, 3, etc., all the way up to infinity. Then you multiply every number in this list by 2, so that you now have 2, 4, 6, etc., all the way up to infinity. The distance between adjacent number in your list has "stretched" (it is now 2 instead of 1), but can you really say that the total extent of all your numbers has "expanded"? You started off with numbers that went up to infinity, and you finished with numbers that went up to infinity. So the total size is the same! If these numbers represent the distances between galaxies in an infinite universe, then it is a good analogy for why the universe does not necessarily expand even though it stretches.
Finally, I should point out that not everything in the universe is "stretching" or "expanding" in the way that the spaces between faraway galaxies stretch. For example, you and I aren't expanding, the Earth isn't expanding, the sun isn't expanding, even the entire Milky Way galaxy isn't expanding. That's because on these relatively small scales, the effect of the universe's stretching is completely overwhelmed by other forces (i.e. the galaxy's gravity, the sun's gravity, the Earth's gravity, and the atomic forces which hold people's bodies together). It is only when we look across far enough distances in the universe that the effect of the universe's stretching becomes noticeable above the effects of local gravity and other forces which tend to hold things together. (That is why, in the analogy of the tape measure I discussed above, the tape measure that you keep in your pocket does not get stretched, while the one that goes between two galaxies does get stretched. I bet some people were wondering about that!)