It's called "The Echo Effect" and also has some scientific backing to it [0]

> we introduce the echo effect, a less researched phenomenon of verbal mimicry, in a real-world setting. Study participants, 330 currency exchange office customers, were assigned into one of three experimental and two control conditions. Careful attention to research design produced results that address issues raised in the mimicry literature and more clearly define the boundaries of verbal mimicry. The results demonstrate that while repetition of words is important in increasing an individual’s tendency to perform prosocial behaviors, the order in which they are repeated back is not; verbal mimicry is more powerful mechanism than dialogue; and, for nonmimicry control conditions, no response produces the same result as a brief response.

[0] https://doi.org/10.1177/0261927X13506906

The actual claim being made is difficult to see on account of the way the article is written, starting with a misleading title. As far as I can tell, the claim is that while people think that taking another person's perspective is helpful, they are not generally as good as they think they are in doing so, because they rely too much on their own intuitions and do not ask enough questions of the other party. I have certainly met more than a few people who find it inconceivable that anyone would see things differently than they do, though I doubt that many of them are among the people who think there is much value in trying to see the other side's point of view.

This is also called active listening. It's a very good, useful skill to have.

Or they just got tired of arguing with you.

So a win/win!