The continuous combat and fish slap are just referring to the timing of the animations and combat resolutions.
In previous games, the attack animation involves a unit attacking its target for one single animation, then returning to their starting hex, then resolving the combat result (damage dealt to both sides)
In civ 7, the combat doesn't resolve immediately. The attacker and defender begin an animation that persists until the end of the combat. So you order unit 1, 2, 3 and 4 all to attack their targets. They all fight at the same time. Then after you've issued all orders, all of them resolve at the same time. What constitutes the end of the combat? I didn't see*. It could be the end of the turn, or maybe once you've given orders to all military units, or maybe each unit that is attached to a single commander. Maybe even something else.
I think it will be harder to read all of the combat text, but it's probably going to be worth it for the spectacle. I'm pretty excited about it.
Edit: Not sure how I missed this: “And then you end turn, and only then do they return to their paths.” <-- So all of the battles will resolve when you end your turn. If you have a lot of battles all over the map, I guess you won't be able to see the damage numbers for them, so I hope they have a combat log or something else to help keep track.