No one likes to be interrupted.
You’re making some argument, or explaining yourself, and maybe you’re not even entirely sure what you think. Through explanation, you’re finally putting words to your thoughts — but it’s barely there. You’re following a barely-visible trail of words, and then… someone throws you off. They inject themselves in the middle of your word-hunt, which now forces you to react to their words, and now you’re lost.
Often, this interjection is entirely unnecessary and irrelevant, which only makes it feel more frustrating.
Everyone who enjoys good arguments has probably experienced this frustration at some point. But I would like to argue that (1) this particular kind of interruption is not the only kind, and (2), that not all interruptions are a bad thing.
It’s a point worth making because the mirror of negative interruptions is the homing in on interruptions by the speaker, as a kind of excuse to shift the conversation and pin moral blame on the listener. Anyone who has said “quit interrupting me!” (myself absolutely included) is probably guilty of this kind of selective focus. Often, the interruption being blamed is not really the bad kind of interruption.
But first, let’s get into distinguishing the good from the bad.
Most people are familiar with the idea of “active listening.” This art involves a number of different skills, but the main idea is that it makes listening a participatory behavior, one in which you are supposed to interject with clarifications, questions, and occasionally to paraphrase what the speaker has said in order to ensure that you have understood them correctly. When one is speaking to an active listener, one seldom actually feels as though they have been interrupted, even though they very often are, because the active listening allows them to feel heard and understood. This permits us to see the problem with interruption, which is not the interruption itself, but the failure of communication being imposed upon the speaker by the listener who cuts them off. If an interruption can happen which facilitates communication and clarification, rather than blocking it, then “interruption” may be permitted — and even encouraged! — for the benefit of better argumentation and communication.
We can think of the affect of the listener in a kind of grid. A listener can be empathic (attempting to understand) or assertive (attempting to defend their own position); in either case, they can be active in their listening or passive, meaning they sit back and listen without interruption:

Although it is tempting to say that “active-empathic is clearly the best,” the reality is that there is probably an appropriate time and a place for all of these affects. When dealing with fanatics, for instance, I have found it is usually best not to engage too much, but to hold your ground (passive-assertive). But when we are conversing with people that we like and respect, the ideal is definitely active-empathic.
Here is where the value of all of this thought really manifests: picture two people who are speaking earnestly and actively with each other. They are likely interrupting each other, not to bulldoze the other, but perhaps to clarify terms and so forth. If one begins to feel as though he is “losing” the argument, the temptation may be to accuse his interlocutor of incessant interruption (“man, I can’t get a thought straight!”), which will either escalate or drive his opposition from active-empathic to passive-assertive.
The shift is from “empathic” to “assertive” (which we might also call “defensive” here) because if the listener’s own perspective is being shut down, why should they listen to what the speaker has to say? They’ll shut up, but they aren’t really “listening” any more, only taking in your argument and quietly scanning it for problems to confirm their own bias and entrench their own position deeper. This is why pointedly asking others to “stop interrupting” often doesn’t help persuasion, and actively hurts the process.
Now, of course, there are people in the active-assertive quadrant who are going to interrupt you, and not in the positive way. In my experience, these people are usually not trying to be dicks or “just win,” but have been recently convinced of the magnificent importance of a single point which dwarfs all others in significance. They have an axe to grind, or a bubble they just can’t see out of. This can be in something as banal as politics, or as niche as gardening.
This can be a difficult personality to take on, but in my own experience, one can get higher success in returning to positive conversation by shifting into active listening, and letting them speak. If their hobby-horse doesn’t seem so great to you, don’t attack it directly (“I just think you’re wrong man”), but instead go up a level: what assumptions does the superiority of their position rest on? Do you share those assumptions? By bringing it up to higher-level subjects, you dodge a lot of the talking-points, and can sometimes bring a fanatic out of their fanaticism (at least for a while) and have an enjoyable conversation.
In both cases, directly confronting other people about interrupting is rarely an effective means of actually having that productive conversation, and is often just a cover for shifting the subject from the matter at hand to the imperfection of the listener.