The part that my brain seems unable to incorporate is the part where two discrete objects can become farther apart and yet neither one of them is moving
But your brain is working just fine. The two objects
are moving,
in relation to each other.
Think of a mid-ocean ridge. Up-welling from the ridge, spreading both left and right, is new ocean floor. So you have a shell on either side of the ridge. The shells are not moving in relation to the part of the ocean floor, the dirt, mud and muck, they are on but, as the up-welling continues and new ocean floor is made the two shells get further apart. A 2-d example. In space-time this happens in 3-d, in every direction at once, a mid-ocean ridge in every speck of space welling up new space in all directions.
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