big bang theory - Why can we still see 10 billion year old galaxies?

We can't see how those galaxies look right now. If a galaxy appears to be 10 billion light years away, that also means that the light took 10 billion years to reach us. It's a bit confusing that "distance" and "time" are sometimes the same thing in astronomy.

So the light which the galaxies emit today (if they still exist) will reach us in another 10 billion years. What we see right now is light that was emitted 10 billion years in the past.

This also means that by looking at the farthest possible distance, we can see light which was created shortly after the big bang. One such source of light is the ubiquitous background radiation which was probably created by the big bang itself. It's easy to see because it fills all the gaps between the celestial objects.

Unfortunately, this also means that we always get an "outdated" view of the universe. If aliens started to blowing stars at the far side of our galaxy, it would take us 100'000 years to see the light.