An idiomatic expression meaning to interrupt someone (literally 'to break the waist of the conversation'), i.e., to cut off the flow of someone's story or explanation.
彼はいつも話の腰を折るので、話し手が困っている。
He always interrupts people, so the speakers are troubled.
重要な説明の途中で話の腰を折らないでください。
Please don't interrupt me in the middle of an important explanation.
会議中、タイミング悪く話の腰を折られてしまった。
During the meeting, my point was awkwardly interrupted.