Direct Answer
The transcript suggests Paul means there is a share of affliction connected to Christ’s body, the church, that believers—like Paul—end up bearing in their own flesh; in that sense, they “fill up” what remains to be suffered in serving the church. Embracing this confronts modern expectations because Paul not only accepts suffering, he “rejoices” in it when it is “for the sake of his body,” shifting focus from personal comfort to the church’s good.