Or to put the question another way…why do galaxies stay together? On the scale of the universe, galactic distances are quite small…but to stars at the end of our galaxy show a red shift (as they w old if the galaxy itself is expanding)?
Most everything at the size of a galaxy cluster and smaller is bound together gravitationally. That is, at that scale gravity’s influence is stronger than the expansive influence of space itself.
Of the other stars in our Galaxy, some show a redshift and some show a blueshift. But this is entirely due to their motions within the Galaxy, and mostly haphazard from one star to the next-- Cosmological effects are completely negligible at this scale.
Galaxies are gravitationally bound. Note that this was the first sign of dark matter. Galaxies don’t appear to have enough visible matter to be gravitationally bound so the existence of a lot of dark matter (more than of visible matter) was inferred.
The expansion of the universe is like the rich getting richer and the poor getting poorer. Gravitationally dense regions like galaxy clusters will get denser with time as the galaxies eventually meld together. Meanwhile, the empty voids between galaxy clusters will get bigger and emptier.
Eventually our local group of galaxies (which will have all coalesced into one big galaxy) will be so far away from any other group of galaxies that the whole rest of the universe will be invisible, redshifted into oblivion, making our big megagalaxy the only thing we’ll be able to see and interact with.
And the final extention of this theory says that eventually there will be a “Big Rip” when all the forces of nature are overwhelmed by the expanding universe and all matter will be torn apart.
What will humanity do when that happens? Is there any escape?
Ultimately, no.
Of course, we’re also talking about time scales so vast they boggle the human mind. In all likelihood, there won’t be homo sapiens running around anymore. If there’s anything at that point descended from us, it’ll be whatever species have evolve from us (and not directly but several grandchild species along the way). If you want to include them as “human”, they also won’t survive. But understand that the time scales involved are so vast, we are more closely related to the primordial ooze than these descendants will be from us.
There’s also the possible chance we’ll develop into a science-fiction type species that has transcended the material universe as we currently understand it, but I was discounting that possibility.
Actually, the Big Rip could plausibly be the quickest way the Universe could end, and could perhaps be as soon as millions (with an M) years from now. Though it should be noted that this is purely speculative: The simplest model of the dark energy would have it constant in time, not increasing, and this is in fact consistent with the error bars in our measurements. You only get a Big Rip if the dark energy increases with time, and nobody has come up with any plausible explanation for how that might happen.
Lots of duct tape.
And I mean lots.
At the origination of the Big Bang, the concentration of Dark Matter that enters our dimension is greatest, pushing everything away from each other ( Red Shift ). Dark Matter surrounds every galaxy keeping it from expanding as it spins like creamer spinning in a coffee in a cup while gravity keeps it in check. Of course this is all just theory.
Don’t forget bailing wire. Lots of it too! 
Ferroll, you appear to be misunderstanding a few key concepts there. First of all, dark energy causes expansion, but dark matter does not: So far as we know, the only connection between the two phenomena is that there’s a lot we don’t know about both. Second, there’s no evidence that either dark matter or dark energy can “enter our dimension”: There are some models of both that could reasonably be described that way, but nobody knows whether those models are correct. Third, dark matter doesn’t surround galaxies: It’s within them and a part of them, in the same way that the visible matter is, and pulls matter in gravitationally to stay with the galaxies in the same way that gravity works for visible matter. Fourth, galactic dark matter is certainly a “theory”, but that word doesn’t mean what you think it means: A theory is the highest level any scientific idea can ever attain, and so it makes no sense to say that something is “just a theory”. No specific model of dark matter or of dark energy is a theory, since all of the models are more tentative than theories.