The #1 pick is much more valuable to most teams than the #3, so I suspect that this deal happened only after they concluded there was not a star trade that was worth pursuing THIS offseason, or not one that they could wait for. Once you draft a player it's very awkward to suddenly deal him a month later (this is what happened to Wiggins), though I'm not saying it's something the Celtics would never consider.
After that, they looked at this draft and decided Fultz was not necessarily the player they wanted most and figured they'd trade down and get another asset. They probably couldn't wait until the draft to make the deal because Fultz needed to work out in Philly and the Celtics needed to run their own workouts with potential #3 picks due to a lot of them, for various reasons, avoiding working out for Boston.
The extra pick will come in handy next year if there is a move to be made for a star, and I think I know who they are dreaming of. It's obviously Anthony Davis who would be the perfect fit. The Pelicans aren't ready to deal him now, but after potentially losing Jrue Holiday and having another disappointing season, the pressure to take Boston's godfather offer will be even greater.
I think the Celtics will just keep on trying to roll the pick over unless they are in a position to draft someone who is a super-duper star. They must have concluded that Fultz isn't on that level. I don't think it had anything to do with Thomas' presence, because Ainge isn't stupid enough to give up on 8-9 years of a James Harden type just because we still have a few more years of Thomas' prime left.