The universe is expanding because the big bang was, well, big, but everything in it has been cruising along just under gravity since then (yes, I’m going to ignore dark energy). So, we have this big explosion in space and stuff comes flying out. Now, a bit of that stuff collapses under gravity and sticks together. Things far away from our new [star, galaxy, cluster] are still moving away from it, but our star doesn’t care. It just keeps happily doing star things while the rest of the universe keeps moving away.
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