This is from The Narrative of W. Spavens, Chatham Pensioner and is describing the way meals are served on board ship. The preceding text describes how the order of calling people for meals is changed around in the interests of fairness, mess shifts are changed regularly and the order of calling is reversed each day. This gives us the clue to understanding the passage in question. It is a method of avoiding favouritism, or its appearance, in serving food.
So this section
"As to the matter of serving, it is this - they give a man liberty to
choose a piece [of meat] for his mess, and then hoodwink him, and when
the Steward calls a mess, he touches a piece, and the Cook give it to
the man it is for, and he touches again."
can be understood to mean:
At each meal one man is chosen. He selects his own portion of meat (which is set aside for him to eat later, or he eats before the rest, the text doesn't specify). He is then blindfolded. In this condition he selects, by touching, the portion for the next man to be served. The Cook serves that portion and the man selecting touches again to indicate the portion for the next man.
