It doesn't mean to see the moon "rising" close. (They didn't see the moon as "rising" at all -- it was always in their field of view and never blocked by the earth.)
It means that they were the first people who ever got near the moon, the first who ever had a chance to look at it from a close range or a short distance.
> "The scientist bent over to see the insect up close."
> "She looked like Sally when she was far away, but when I saw her up close, I saw that she was not really a lookalike."
> "I've never seen a shark up close -- luckily!"
> "There were so many people at the rock concert that I didn't get a chance to see Mick Jagger up close."