Our Mishnah lists a series of cases in which the Sages were lenient and accepted a person's testimony about things he saw when he was a minor. Today he is already an adult, but the event he is testifying about he witnessed as a child. Ordinarily such testimony is not accepted, but in these cases, since there is room to be lenient, the Sages permitted relying on it.
The words of the Mishnah:
"V'elu ne'emanin l'ha'id b'godlan mah shera'u b'katnan" - these are the situations in which a person is believed to testify as an adult about what he saw as a child.
Testimony about handwriting:
"Zeh kesav yado shel abba" - a person is believed to say that this is his father's handwriting, even though he only saw his father's signature as a child.
"Zeh kesav yado shel rabi" - and likewise regarding his teacher's signature, which too he only saw in his childhood.
"Zeh kesav yado shel achi" - and likewise regarding his brother's signature.
The reason for the leniency: it is the Sages who instituted the concern that a document bearing signatures might be forged, and therefore the signatures must be verified - this is what is called 'kiyum shtaros' (authentication of documents). Since this is only a rabbinic requirement, the Sages were lenient and ruled that one may rely on what the witness saw in his childhood.
The Gemara adds one point to this: one who saw the matter as a minor is believed only as the second witness, but one fully valid witness who saw the matter as an adult is still required.
Testimony about the hinuma:
A person is believed to say "I remember" - that he recalls "shep'lonis sheyatz'ah b'hinuma v'roshah paru'a": on her wedding day she went out in a hinuma, a veil worn only by virgins, and her hair was loose over her shoulders, which is likewise a sign practiced only by virgins. On the strength of this testimony it can be established that she married as a virgin, and her kesubah is two hundred zuz, as we learned at the beginning of the chapter.
Signs of priesthood:
They are also believed to testify "v'shehayah ish p'loni yotzei mibeis hasefer litbol le'echol b'Terumah" - that they saw him leaving the schoolhouse where Torah was taught, immersing, and going to eat Terumah. Even a child may eat Terumah, and if it can be established that this person acted this way, that serves as proof that he is a kohen.
The Gemara asks: how do we know that he is a kohen? Perhaps he is a kohen's slave, since a kohen's slave also eats Terumah, and it was for that purpose that he went down to the mikvah. Regarding this we learned that he was leaving the schoolhouse: he was coming from Torah study, and a slave may not study Torah, and from here we may assume that he is not a slave.
One further clarification: even though he is permitted to eat Terumah as a kohen, this applies only to rabbinic Terumah, for all of these leniencies are stated only regarding matters that are rabbinic in nature.
Another sign concerning which they are believed on the basis of what they saw in their childhood: "v'shehayah cholek imanu al hagoren" - that he was a kohen who took his portion at the threshing floor as a kohen. This testimony too applies only to matters that are rabbinic in nature.
Beis haperas and the Shabbos boundary:
"Bamakom hazeh beis haperas" - they are believed to testify that they knew in their childhood that this field is a beis haperas, that is, a field considered impure by rabbinic law. One cause for this is a grave that was in the field and the field was plowed, so we are concerned that fragments of bone may have been plowed up and scattered in it; therefore the Sages said that such a field is considered impure. And just as he is believed to say that this field is a beis haperas, so too he is believed to say that another field is not a beis haperas, even though he saw this only as a child.
"V'ad kan hayu ba'im b'Shabbos" - that up to this point people would regularly come on Shabbos, because it is within the boundary, two thousand amos from the city. The measure of two thousand amos is also rabbinic law, and therefore he is believed regarding it.
The matters concerning which they are not believed:
However, the Mishnah says that a person is not believed to say, on the basis of what he remembers from his childhood, that a road or a path in his fellow's field belongs to a particular person, and likewise he is not believed to say that a place of ma'amad or misped belongs to a particular person. The 'ma'amad' and the 'misped' were places where people stopped along the way while bringing a corpse for burial: an area called ma'amad, and an area called misped, where the eulogies were delivered. These testimonies are not accepted because they are monetary matters, whose standing is Torah law, since through such testimony money is taken from one person's possession and given to another. In these matters a person is believed only about what he saw as an adult.
In summary: the Mishnah lists the cases in which the Sages were lenient and accepted testimony about what the witness saw as a minor: identifying the handwriting of a father, a teacher and a brother for the purpose of authenticating documents; the testimony about the hinuma and the loose hair, which establishes a kesubah of two hundred zuz; the signs of priesthood - leaving the schoolhouse to immerse and eat Terumah, and taking a portion at the threshing floor; and also beis haperas and the Shabbos boundary. In all of these the leniency stems from the fact that the matter is only rabbinic, and in the case of handwriting testimony it is further required that the second witness be fully valid and have seen the matter as an adult. But in monetary matters, which are governed by Torah law, only testimony about what he saw as an adult is accepted.