Maseches Ma'asros, Chapter 2, Mishnah 1 - a fairly complex mishnah. Its central point is simple: a person who is carrying his produce home and offers some of it to passersby along the way - since he has not yet brought it into his house, the produce has not reached the completion of its processing and is therefore exempt from tithes. The recipient may accordingly eat from it without separating anything, since this is a casual eating. But once the produce has been brought into the buyer's house - it is his food and it is in his house - he becomes obligated to tithe it properly. That is the main point of the mishnah.
The secondary topic: the am ha'aretz and demai
Alongside this, the mishnah introduces a further matter, with something of an ironic twist: when the source of the food is an am ha'aretz - a person who is not careful to separate tithes in their proper order - the one receiving from him must be concerned that the produce may not have been properly tithed. This is in fact the subject of Maseches Demai, where the matter is explained at length.
By way of a brief review: in the days of the Second Temple a survey was conducted at the initiative of Yochanan the Kohen Gadol, and it emerged that most of the amei ha'aretz did indeed tithe their crops properly - they separated Terumah Gedolah, Ma'aser Rishon and Ma'aser Sheni; but a significant minority among them did not separate Ma'aser Rishon and Ma'aser Sheni.
Ma'aser Rishon: the gift given to the Levi, a tenth of the crop in the hands of the Yisrael.
Terumas Ma'aser: the gift the Levi gives to the kohen - a tenth of the tenth, that is, a tenth of what the Levi received.
Included within Ma'aser Rishon, then, is Terumas Ma'aser, and eating it forbidden carries the severe penalty of death at the hands of Heaven.
For this reason the Sages instituted a series of rules known as demai: one who buys produce from an am ha'aretz separates from it the Terumas Ma'aser that may be required, and redeems onto a coin the Ma'aser Sheni that may be required.
The case the mishnah is dealing with:
The mishnah deals with a person returning home from the field, from the place where the food was prepared, so that his produce has not yet reached the completion of its processing, and he offers some of it to passersby. If he is an am ha'aretz - this does not present an obstacle, and since there was no completion of processing, one may eat from the produce as a casual eating. But if he said that he had already tithed it, he can no longer be relied upon to have tithed it properly: although one may safely assume that Terumah Gedolah was separated, there is reason for concern about Terumas Ma'aser and Ma'aser Sheni.
An important note in understanding the wording of the mishnah, "hayah over ba'shuk" - he was passing through the marketplace: this does not mean that he is walking through the marketplace in order to sell his produce, for in the case of someone who intends to sell, the completion of processing is established already when he finishes his work in the field. Rather, it refers to someone carrying his produce from the field to his house, and therefore the completion of processing has not yet taken effect, since the produce has not reached his house; and he is passing through the public domain, which is called here a marketplace.
The first case: "tlu lachem te'einim":
The owner of the produce says to his friends: "Take some of my new figs," out of his joy at having brought them in from the harvest. Since the produce has not yet reached the house, it is exempt from tithes, and therefore - even though the one offering it is an am ha'aretz - his friends may take from the figs and eat them casually in the marketplace, and they are exempt from any tithe.
The flip side of this rule is equally true: if these passersby took the figs home with them, then once a fig that has become theirs enters their house it becomes subject to tithes, and they must tithe it properly. The Mishnah's term "vadai" - certainly - means that these fruits are not treated as demai, but rather as food that certainly was not tithed: from the very fact that the am ha'aretz offered them to every passerby to take, everyone understands that the fruits are untithed, only that they had not yet become obligated in tithing. And since they were not tithed beforehand, one who brings them into his house must tithe them as certainly untithed produce, separating from them Terumah Gedolah and all the other gifts, since there is no claim at all that they were already tithed.
The second case: "Telu lachem te'einim me'usaros" - "Take some tithed figs for yourselves":
If the am ha'aretz says as he passes through the marketplace, "Take some of my figs, and you may even bring them into your homes, for they have already been tithed" - he is believed regarding his claim that the fruits already became obligated in tithes and that he began tithing them, that is, that he separated Terumah Gedolah and the like; but he is not relied upon to have properly separated terumas ma'aser and ma'aser sheni. For this reason it is not permitted to eat these figs even as a casual snack in the marketplace, since we must be concerned that the obligation to separate terumas ma'aser and the like still rests upon them.
And when a person brings these figs into his house, he does not recite a berachah on the separation of Terumah Gedolah, because the am ha'aretz said that they were already tithed, or at least hinted at it. Yet he is still required to tithe them a second time, lest they were not tithed properly the first time, and therefore he must separate terumas ma'aser and ma'aser sheni before eating this food, just as is the law for any demai produce purchased from an am ha'aretz.
In summary: in this Mishnah we learned that fruits being brought from the field to the house have not yet reached the completion of their processing, and therefore when someone offers them along the way - even if he is an am ha'aretz - they may be eaten as a casual snack in the marketplace; but one who brings them into his house becomes obligated to separate the tithes. When the one offering them made no mention of tithing at all, the fruits are tithed as certainly untithed produce; and when he said that they are tithed, they may no longer be eaten as a casual snack, and when they are brought into the house they are tithed according to the law of demai - by separating terumas ma'aser and ma'aser sheni, and without a berachah.