Let me check but I think it refers to a shared dish; at an izakaiya you often order a bunch of shared food plates and then serve yourself from them.

It is definitely rude to use chopsticks that you just put in your mouth to go rooting around for something in those. You are supposed to take from the top and ideally turn them around using the back end. Some people frown on using the back ends however as it may have been touched by your hand...

Edit add: It means to dig food out, either from your own dish or a shared one. Like mixing the food up to look for something you like in it.

I think just written in an ambiguous way: "dish" here refers to the food contained in the vessel and not the vessel itself.