There is actually an interesting issue here. Our cosmological models are going to employ the "cosmological principle" (the attribute that the universe is homogeneous and isotropic at large enough scales, so looks the same everywhere but shows change over a meaningful universal time coordinate) for as long as the evidence allows that principle to be used (which it currently does), because that is a vastly unifying and simplifying principle that makes cosmology possible. So think about that principle for a moment-- everywhere is the same, the universe looks the same from any point and from all directions, at any given age. This automatically implies that every point will seem like the center to an observer at that point.
Thus, the real question to ask is, why does the universe obey the cosmological principle? It is a symmetry principle, so we are asking, why does the universe respect that symmetry, instead of shattering it? These are hard questions to answer, because we normally view symmetries as special and less likely. I see something unanswered behind the question "why does the Earth appear to be at the center of the universe?", but the question in no way picks the Earth out as special. Instead, it picks the universe out as special, not our place in it. We may be asking the wrong question.
