Hi
I'm not sure about the answer, but as far as I know, "to bug" means to record someone's moves or sayings and to know what they're doing or talking about. Another meaning for bug can be being irritated or annoyed about something which I guess is the best answer for this case.
Being bugged by news means that you hear a lot of bad news from your environment that you feel annoyed and depressed and irritated.
(I hope that helps!)