I just sent one of these, and I think it can be pretty simple, with varying degrees of detail depending on how much interaction you’ve had with the school. The format that I followed was: 1. Thanking them for their interest in me and offer of admission 2. Letting them know that I appreciated the interactions I had with faculty members (i.e. my interview), and 3. Letting them know I had decided not to accept the offer.
The response I got was very positive, I think they were thankful I got back to them early because that lets them offer a spot to someone else. Don’t worry too much about it, all departments understand that most applicants apply to multiple schools, so they are prepared to receive these emails. One thing I would advise against is telling them which school’s offer you plan to accept (just not necessary). I hope this helps
And congrats to all the admits! I haven’t heard from Princeton but I wouldn’t be too heartbroken cause I have other options.