Simple experiment: Blow up a balloon half way but do not tie it off, just pinch the end. This represents the universe. Now take a marker and put dots all over the surface of the balloon. These represent stars, planets, everything. Now start blowing into the balloon again to simulate the universe's continued expansion. You'll notice that all the dots are moving away from each other on all sides. No dot is getting closer to any other dot. This would also be true if you could somehow place dots inside the space of the balloon; and while it would remain true for a dot in the very center of the balloon, it also remains true for every other dot.
So seeing everything moving away from us does not require us to be at the center. We're just another dot...a pale blue one.