Lois sees a video where Oliver tried to commit a suicide.
Lois: I saw the video. That's enough to send anyone
on a binge.
Make you sad? Why didn't you come talk to me?
It means to cause someone to become emotionally upset and seek solace in some form of excessive behavior.
It would usually be drinking, but it could be anything - a murder rampage, an eating binge, a gambling binge, sex, drugs, rock'n'roll - you name it!

We also use "binge" as an adjective: "binge eating," eg., as with bulimics.
Thank you, Avangi!