"Left off" means "stopped" or "paused", and is generally used in the context of resuming the activity that had been paused.
There's a bit of subtlety in the first example, though. The quote does not mean to imply that talk about #MeToo literally stopped after the Golden Globes and didn't resume until the Oscars. The author is trying to emphasize the continuity between the talk at the two events, while implying that there haven't been any major changes in what people are saying in the meantime.
The second example is more straightforward. The people involved were doing something (probably having a conversation, but not necessarily), which ended unresolved when the speaker fled the apartment.