If space increased faster than light moments after the Big Bang, why do we see anything near us in space at all?

Because space dilation does not occur at EVERY scale. Local matter can provide a stronger local gravitational field than the rest of the universe so that the local dynamics of space are not affected by the global dilation. We see this effect even today because the cosmological expansion only 'wins' at scales of about five million light years or larger. For smaller regions, the local gravitational field is what's important.

Copyright 1997 Dr. Sten Odenwald