What is the expanding universe expanding into?

We know our Universe is expanding. What does it expand into? If you traveled to the very edge of space-time and stuck your hand out of it, what would you be sticking your hand into? Or could you travel forever and never get to the edge of it, even though it is finite? Is it like the train station Neo was stuck in in the Matrix? See Question 3 for more on Neo and the Trainman’s train station.

Inquiring minds want to know. What’s really out there and who put it there?

A similar question was asked over a year ago, but I hope this one is different enough to provoke more discussion and clarify what has been learned in the last 18 months. Also, I thought it worth including because it fits in a growing series of questions about our Universe. There has been:
1—Big Bang Theory—How can you divide infinity into a single finite whole?
2—How would you answer this speed-of-light question?
3—What happens when the expansion of the Universe reaches the speed of light?