This is a quite common construction that can be used with about any verb that describes an activity. The idea is that you go to another place in order to do the activity there, along the lines of "to go out to do X" or "to go somewhere to do X there". The place you go to is often specifically designated for the activity. For example
Wir gehen am Mittwoch essen.
We go out to eat on Wednesday.
means that you go somewhere, being to the restaurant, in order to eat there. The restaurant is meant to be a place where people eat.
Other examples might be
Ich bin müde, ich gehe jetzt schlafen.
I'm tired, I'll go to bed now.
meaning that you go from where you are, for example from the living room, to the bedroom and sleep there. The bed is meant for sleeping.
Ich muss noch einkaufen gehen.
I still have to go shopping.
You go to the store, which is meant for people to shop there.
Möchtest Du spazieren gehen?
Would you like to go for a walk?
Here the designation of the place is less clear, because you can go for a walk just about anywhere, but you still go to another place than you are right now.
Meine Tochter geht oft Fußball spielen.
My daughter often plays soccer (at the soccer pitch).
Du musst heute nicht arbeiten gehen.
You don't have to go to work today.
A similar construction is used in cases where there's no physical movement, but more a change of state:
Frank geht in Rente.
Frank retires (goes into retirement).
Die neue Anlage geht in Betrieb.
The new facility goes on line (goes into operation).
"Essen gehen" specifically typically describes going to professional establishment like a restaurant. If you visit your brother and have dinner with his family, you probably wouldn't call it "essen gehen".