“piece of bread.” Abraham got more than a “piece of bread.” It was a biblical custom to take excellent care of guests and feed them well, both as a blessing to them and as a sign that God had blessed your house with all you needed. This custom was why Jesus could tell the parable of the man who had a guest come but had no bread, and so made the effort to wake up a neighbor even though it was late at night (Luke 11:5-10). But you must take care of your guest without making it seem like you are going to any trouble, even though you are. So Abraham, acting like it was no problem, told his guests to please take time to rest, wash their feet, and he would get them a piece of bread to eat.
Abraham would have taken good care of any guest. But in this case, Abraham knew he was feeding God, who had come to his house in human form, so he wanted to take especially good care of Him. He had Sarah get 3 seahs (about 21 quarts, or over 5 gallons [22 liters]) of fine flour for bread, and biblical “bread” was flatbread, like a pita or pancake. It usually takes about ¼ cup of flour to make a good-sized pancake, so at ¼ cup per flatbread, Sarah could have made over 250 loaves of bread with the amount of flour Abraham said to get (a full flatbread is referred to as a “loaf” in many Bible versions).
Then Abraham selected a tender young calf (the Hebrew uses an idiom and reads, “a son of the herd”) and hurried to prepare it. Generally, that preparation would have been to butcher the calf and then boil it, making a kind of stew that could then be eaten using pieces of the bread as spoons. People did not use forks and spoons as eating utensils in the biblical world of the Old Testament. By Roman times, the common people of Israel would have maintained the ancient custom of eating with the hands, using bread as a spoon, but many of the Romans used at least some utensils to eat. The spoon was the most prevalent utensil, then the knife, then, and rarely, a kind of fork (most of the time, if meat needed to be stabbed, the knife would do double duty).
Abraham had made it seem to his guests that feeding them was no problem to him and no inconvenience to the guests: just a piece of bread, a little rest, and they could be on their way. In reality, things were much different (and usually both parties understood that). Making the huge amount of bread would have taken some time; as did killing, butchering, and boiling the calf. No doubt at least a couple hours had gone by before Abraham was ready to set the feast before them, and then he acted like a household servant and stood watching over their needs while they ate, ready to pass them what they needed, get more of anything that needed to be replenished, and pour water over their hands when they were done to cleanse their hands (cp. 2 Kings 3:11).
Having a host stand and wait on the “table,” which for tent-dwellers was usually a cloth spread on the ground, while you, the guest, ate, would make any modern guest uncomfortable, but the people of the time understood the special treatment that guests received, and so there was no protest from the three guests when Abraham stood and watched as they ate.
“for that is why you have come to your servant.” As a part of his hospitality, Abraham makes it seem like the reason that the three men have come that way is so that they can honor Abraham by letting him take care of them.