Interrupting tells people your next thought matters more than their current one.

Cutting in mid sentence delivers a message you did not intend to send, which is that whatever arrived in your head outranks what is coming out of theirs. Most interrupting is not rudeness at all. It is enthusiasm, or fear of losing the thought. The other person cannot see the reason, though, and only feels the door close.

Park the thought instead of spending it immediately. If you are worried about losing it, write three words down. Then let them land the sentence, and whatever you were going to say will be better for having heard the ending.