Do I need to explain my reasoning when choosing between two offers I have been lucky enough to have been offered different professor positions in different universities. Both departments have made a lot of efforts to try and convince me to choose theirs (who gets offered what job is public here).

It was not easy, but I have finally chosen one. Should I try to explain why to the people from the other department, the one I am not going to? This is not obvious, because while there are arguments I could make, my decision relies in large part on extra-academic concerns, and some gut feeling. However, such an explanation may be expected, and I do not want to burn any bridges.
No.  You do not need to explain your decision.

Assuming it's actually true, it would be appropriate to tell the department you're turning down that their offer was attractive, and that it was a difficult decision, and that the choice came down to non-academic factors that were out of their control.  But even this is more than you need to tell them.