Human language simply has not evolved to properly frame these ideas. The universe has become cooler and less dense as time has progressed. From any local point of view, all other points appear to be moving away. The best description of this phenomenon is “expansion”.
Naturally, we want to ask “Expanding into what?”, since all of our mundane experience with expansion involves expansion into the surrounding space (or spacetime, if you prefer). Our minds conjure up a context.
Personally, I believe our universe does indeed have a hyperdimensional context within which its development can be said to take place. However, we will only be able to detect this context indirectly. Hell, we now understand that we can’t even detect the full scope of our own universe directly – a fact which saddens me profoundly.
Unfortunately, as someone mentioned above, even if we are able to come up with verifiable theories that describe the Big Bang, it only removes the problem another step. No matter what we do, our species is doomed to perish only knowing that “it’s elephants all the way down”. The fundamentally unanswerable question is: “Why is there anything, instead of nothing?”
But as to your specific question, imagine the collision of two hyperdimensional entities. Their intersection might spawn a 4-D (or 11-D for you string theorists out there) dynamic universe, just as the collision of 2 planes would spawn a dynamic line which changes and evolves as the planes move in relationship to one another.
Or imagine a hyperdimensional space which undergoes a rapid expansion, like carbonated water when its container is uncapped. Bubbles form, expand, and burst on the surface or break into smaller bubbles. Our universe may be very similar to such a bubble.
I would be surprised if our universe actually can be accurately described as a Euclidian shape. My guess is that it’s probably a much fuzzier, messier thing.