I've been on the other side of this. An interviewee had a killer competing offer that I couldn't match. He was so apologetic, as if he was worried he'd insulted me. I had to assure him that I took no offense and wished him well. Don't know why anyone would take something like this personally.
I've been that interviewee before and I was much the same. For me I realized it has to do with viewing the time spent in the interview as some kind of unspoken contract. You both felt good about the interview but you aren't accepting. Feels like you're letting them down.