Basically, they said the effect from Angel's Grace should wear off and then stop at the cleanup step if there is a trigger in the cleanup step, which then allows the active player to do things in response to the trigger. But it didn't work. But is that supposed to work like that?
Or is Magic Online bugged? I mean even the most sportsmanlike opponent will not concede unless they know what's happening!
Yes, they should have died. There are two things Angel's Grace is doing: Stops Damage from putting your life below 1 "until End of Turn" and prevents you from losing "this turn". So, the damage from Borborygmos, since it is being done in the Cleanup Step, would put the opponent to negative life. This is because the Until End of Turn effect wears off as soon as the Cleanup Step starts so it is no longer in effect at this time. And the "this turn" effect wears off as well even though it is in the same turn.
514.2. Second, the following actions happen simultaneously: all damage marked on permanents (including phased-out permanents) is removed and all "until end of turn" and "this turn" effects end. This turn-based action doesn't use the stack.
So, it does appear that MTGO is bugged in this situation.
For convenience, explain what is happening in your thread, rather than pointing to a video for this.
It appears that Worldspine Wurm is being discarded as the cleanup step begins, making its last ability trigger during the cleanup step (C.R. 514.1, 603.2). After that, "this turn" and "until end of turn" effects, such as those of Angel's Grace, end (among other things) (C.R. 514.2). Then the active player should have gotten priority (since a "triggered abilit[y] [is] waiting to be put onto the stack") (C.R. 514.3a); if not, then that is a bug in Magic Online.
https://www.youtube.com/watch?v=SqdztFWhbf4
Around the 19min-30sec mark.
The chat said the opponent should have died.
Basically, they said the effect from Angel's Grace should wear off and then stop at the cleanup step if there is a trigger in the cleanup step, which then allows the active player to do things in response to the trigger. But it didn't work. But is that supposed to work like that?
Or is Magic Online bugged? I mean even the most sportsmanlike opponent will not concede unless they know what's happening!
So, it does appear that MTGO is bugged in this situation.
It appears that Worldspine Wurm is being discarded as the cleanup step begins, making its last ability trigger during the cleanup step (C.R. 514.1, 603.2). After that, "this turn" and "until end of turn" effects, such as those of Angel's Grace, end (among other things) (C.R. 514.2). Then the active player should have gotten priority (since a "triggered abilit[y] [is] waiting to be put onto the stack") (C.R. 514.3a); if not, then that is a bug in Magic Online.Borborygmos has no relevant abilities for dealing damage in the cleanup step. You probably mean Borborygmos Enraged.
EDIT (Apr. 11, 2020): Struck out some text.