to milk X

This expression comes from farming and milking a cow. Yes in English milk can be used as a verb!

The original meaning is simply: to milk a cow. 



From this verb we got the expression to milk something for all it's worth. 

Hal: Did you hear that Tomo won an award?

Bill: Yeah, he's milking it for all it's worth! He's been telling everyone he meets about it.

This was eventually shortened to milk X. So this usage of milk means to exploit something as much as possible, to take of advantage of something as much as you can.

Husband: Oh, I'm so tired from yesterday's marathon. Can you do the dishes?

Wife: Stop milking that! Yes you ran a marathon but you haven't done any work since then!

Fred: He's really milking the sympathy card from his breakup.

Jeb: Yep, he's had a whole string of new ladies recently.

Fred: Well wouldn't you do the same thing?

Jeb: Yeah, probably. hahaha

Have a nice weekend,

Phil