Will see the universe a bit differently. There are a few things that make this point in time very special. Most people know the universe is expanding. This is slightly misleading, however. Space itself is expanding. Alex Filippenko, professor of Astronomy at Berkley, explains it with the best analogy. Imagine ping pong balls side by side on a stretchy tube of rubber. Then if you pull apart the tube, the ping pong balls separate from each other. The ones in the center separate the least (this is because the center is expanded the least). It's a really good analogy for how dark energy is pushing the objects further away from us further away even faster.
The other big thing that makes living now very unique is the afterglow radiation of the big bang. About 3% of it remains, and it has been mapped with a microwave telescope a little while back. That radiation is the only other evidence a big bang ever existed. Civilizations that spawn several billion years from now will see no other galaxies in the sky, and no afterglow from the big bang. The only thing they will know exists is the Milky Way. Lawrence Krauss, theoretical physicist from Arizona State, says no evidence at all will exist of these other hundreds of billions of galaxies. Sad isn't it?
It makes me wonder if we missed some event 2 billion years ago that would have made it clear how the universe started.