When we are interrupted we feel short-changed, disrespected, unheard, and less important in the interrupter's mind than they are. When someone lets us speak without interruption and even gives a little space at the end of what we are saying to make sure we are done, and maybe even asks a question to have us tell them more, we feel really appreciated, validated and heard.
So why do we interrupt? Sometimes we are impatient. Sometimes we think we know what they are going to say and our mouths just blurt out what the rest of the sentence (which is often wrong) without a filter to stop it. Sometimes we are sick of hearing someone and want them to stop talking. Sometimes we want someone to hear us so badly we can't wait through what they have to say, and so tell them in the middle of their sentence what we are thinking about, however unrelated it might be. And sometimes we weren't really listening to someone, were in our own thoughts and just said the next thing that occurred to us to say. And sometimes we are excited to share commonality with another person, to show how much we are alike, so we interrupt everything they say that triggers a similarity we have.
We interrupt either because we are excited about someone, not excited at all, not thinking, or are actually feeling negative about someone. The common denominator in each case is that we are consciously or unconsciously trying to show that we are as important or as interesting, if not more so, than another person. Period.
And the person who has been interrupted feels this. Don't you when you are interrupted?
In order not to interrupt it often takes a making a conscious intention before or during a conversation to give the other person the floor, to let them be the most important or interesting person in that moment, or for quite a few moments. The interesting thing is, when we let someone finish what they have to say, they usually then ask us to share something about us. Then we get a chance to be important and interesting, and respected as well, just because we didn't interrupt.