Refers to the common phrases one may hear from people when they tell others they do not want kids. Imagine it like a BINGO board, covered in these common phrases. The joke is that the phrases and pleas are so common, it's almost like you can play a game of BINGO with them.
This phrasing is used in the childfree community to outline the stereotypical criticisms people face when they mention they do not want children.
I knew that I didn't want to have kids since I was a teen, but every so often when the topic comes up with a date, she will bingo me.
Last night, my parents bingo'd me about when they'll be having grand children. They don't know about the vasectomy, and neither does my girlfriend.