We are learning Mishnah 7 in the eighth chapter of Tractate Berachos. The Mishnah deals with someone who finished his meal and forgot to recite Birkas HaMazon: "mi she'achal v'shachach v'lo beirach" - one who ate and forgot and did not recite the berachah. Although the Mishnah does not spell this out, it means that the person has already left the place where he ate - the place where the obligation to recite the berachah fell upon him.
On this point Beis Shammai and Beis Hillel disagree:
"Beis Shammai omrim: yachzor limkomo vivarech" - Beis Shammai say that he must return to the place where he ate and recite the berachah there.
"U'Beis Hillel omrim: yevarech b'makom shenizkar" - Beis Hillel say that he may recite Birkas HaMazon in whatever place he remembered.
The parameters of the dispute:
The Gemara emphasizes that the dispute between Beis Shammai and Beis Hillel was stated specifically regarding one who "forgot" - where he genuinely forgot to recite the berachah, and his departure from the place was unintentional. But someone who left his place deliberately, saying, "I will go elsewhere and recite the berachah there," is obligated to return to his place and recite it there, and the matter is not left to his choice.
The continuation of the discussion in the Beraisa:
The Beraisa records the continuation of the exchange between Beis Hillel and Beis Shammai. Beis Hillel said: if a person ate at the top of a tall tower and forgot to recite the berachah, and after descending he remembered - would you really obligate him to climb all those stairs back up in order to recite the berachah? Beis Shammai answered: according to your reasoning, if a person forgot his wallet and his money at the top of that same tower, would he not go back up to retrieve it? And if he climbs back up for his own needs, how much more so for the needs of Heaven.
"Ad eimasai mevarech?":
The Mishnah then asks a further question, which likewise concerns fulfilling the obligation of Birkas HaMazon after the fact: "ad eimasai mevarech?" - until when may he recite the berachah? This is a common and practical question: someone who forgot to recite the berachah, until what point after eating does the possibility of reciting it remain open to him? The Mishnah's answer: "ad k'dei sheyis'akel hamazon shebimei'av" - until the food in his intestines is digested, that is, until the food has been digested in the digestive system.
And how can one know the measure of this time? The Gemara provides a sign: as long as he is not hungry. If, on account of the meal he ate, the person is still not hungry, this is a sign that the food has not been digested, and therefore he is still obligated in Birkas HaMazon.
Now, the digestive process continues even after that point, but Rabbeinu Yonah explains that from the moment a person begins to digest the food he is liable to feel a certain degree of hunger; and once a slight hunger stirs within him, this is an indication that the food is considered fully digested, and he no longer recites Birkas HaMazon.
In summary: in this Mishnah we learned two laws concerning Birkas HaMazon after the fact. As to the place of the berachah: the dispute between Beis Shammai ("yachzor limkomo vivarech") and Beis Hillel ("yevarech b'makom shenizkar"), which applies specifically where he forgot unintentionally, along with the kal vachomer that Beis Shammai offered in the Beraisa, from a wallet to the needs of Heaven. And as to the time of the berachah: "ad k'dei sheyis'akel hamazon shebimei'av", whose measure is as long as he is not hungry.