According to the effects of general relativity, this is indeed an "illusion". We do appear to be at the center, but any observer anywhere in the universe will appear to be at the center, from their own perspective.
It's sort of like being in an old video game (think back to the old Atari 2600 games) where if your spaceship flies off one edge of the screen, it comes back in on the other side. This is how the general relativity model of the universe works - except that in the real universe, the further away you move from the 'center', the faster objects recede, until the 'edges' (of the 'screen', if you will) are receding away from the observer at the speed of light. Because we can never reach these edges (of the event horizon, it's called), the paradox of flying off one edge of the universe and coming back in on the other side can never occur. ('Looking' at this edge, about 13.5 billion light years away, (which we can't quite see due to the infinite redshift at this distance) is, in essence, looking at the moment of the Big Bang.) If you could travel faster than light, in principle, you could travel out one edge of the universe and emerge on the opposite 'side' (which, one could say, is part of the very reason it's impossible).
Another way to look at it is to imagine being on the surface of a giant balloon that is inflating at a speed so fast that if you try to travel on its surface at maximum speed that you don't ever make it around to the opposite side. Anyone anywhere on the balloon will think the opposite side is an "edge" they can't reach. This all sounds strange, but this model of the universe is consistent with Einstein's relativity equations, the cosmic redshift and a homogeneous universe. (I find it fascinating, that even in principle we can't see far enough in space to see what happened at the moment of Creation.)
What do you think of the inflationary universe? It works out to a total universe as much bigger than the Hubble volume as earth is to a grain of sand. One good thing about that model is that it explains why the universe appears so flat, isotropic, homegeneous if we can see only a tiny piece of the whole, which would be curved back on itself.
My brain hurts.
Thanks for the exceptionally clear eplanantion.