We are learning the third Mishnah in the first chapter of Maseches Beitzah. In the times of Chazal it was common to eat pigeons, much as we eat chicken today, and the pigeons were kept in a dovecote - a structure built to house birds, with a small opening for them to enter and many compartments inside.
The Dovecote Ladder on Yom Tov:
The Mishnah opens with a discussion about a person who wants to reach his dovecote by means of a ladder, and the question of which ladder may be used on Yom Tov. After all, carrying is permitted on Yom Tov; the concern here is different: an ordinary ladder is generally used for cleaning gutters, repairing the roof, and other tasks that are not permitted on Yom Tov, and an onlooker may think that the person is taking it for a forbidden task. This concern is the subject of the dispute of the Tannaim before us.
"Beis Shammai omrim: ein molichin es hasulam mishovach leshovach" - one may not carry a ladder from one dovecote to another, out of concern that he will appear to be about to use it for a purpose that is not permitted on Yom Tov. "Aval mateihu michalon lechalon" - but he may tilt it from one opening to another within that same dovecote, in order to reach other pigeons.
"UBeis Hillel matir" - Beis Hillel permit carrying the ladder from one dovecote to another.
The Gemara distinguishes between types of ladders: a ladder designed for going up to the second floor of a house, and not for use with a dovecote - here even Beis Hillel agree that it may not be moved from dovecote to dovecote on Yom Tov, in accordance with the view of Beis Shammai. The dispute concerns only a ladder designed for a dovecote, and it is regarding such a ladder that Beis Shammai are stringent and forbid using it on Yom Tov.
According to the Bartenura and the Rambam, even Beis Hillel agree that in a public domain it is forbidden to move even a dovecote ladder from place to place, for the very same reason: people will say that he is going to repair his roof with it, and even though the ladder is designed for the dovecote, the concern remains in place. The leniency applies only in a private setting, where onlookers are not to be found.
Now, we have a general rule that anything forbidden because of maris ayin is forbidden even in the innermost chambers and in private. Here, the Rambam teaches, the matter is an exception to that rule: because of the joy of Yom Tov, and so that a person will be able to enjoy Yom Tov and obtain the birds he needs, they permitted a dovecote ladder in a private place where no one sees him. But in public, even Beis Hillel would agree that there is a problem here.
Taking the Pigeons on Yom Tov:
From here the Mishnah moves to the question of how and under what circumstances one may take the pigeons on Yom Tov, and whether advance preparation from Erev Yom Tov is required:
"Beis Shammai omrim: lo yitol ela im kein ni'anea" - one may not take pigeons on Yom Tov in order to slaughter them unless he actually moved them and took hold of them on Erev Yom Tov, thereby deciding that it is specifically these that he wants.
"UBeis Hillel omrim: omeid ve'omer zeh vezeh ani notel" - it is enough for him to stand on Erev Yom Tov and say, "This one and this one I am taking," identifying verbally the pigeons that he wants. He need not take hold of them physically.
The Gemara explains that the concern of Beis Shammai, which requires actually taking hold of the birds, applies only to the first brood in the dovecote - the first pair of pigeons taken from there; but with those that grow there afterward, there is no such concern. This is because the common tendency was to leave the first birds in the dovecote, because of the benefit in doing so, and therefore Beis Shammai were concerned that when he comes to take them on Yom Tov he will change his mind and say, "I would rather leave them where they are."
Accordingly, if he only identified them verbally on Erev Yom Tov and did not actually take hold of them, Beis Shammai were concerned that he might change his mind and decide not to slaughter them, and it would turn out that he moved them for no purpose - and one may move things only for a purpose. But if he took hold of them and moved them on Erev Yom Tov, he will no longer change his mind. Regarding the second brood, however, the group that grows there afterward, even Beis Shammai agree that there is no concern that he will change his mind, and therefore verbal identification alone suffices and there is no need to take hold of them.
In summary: in this Mishnah we learned two disputes between Beis Shammai and Beis Hillel in the laws of Yom Tov: carrying a ladder from one dovecote to another, which was forbidden because of maris ayin and permitted by Beis Hillel with a ladder designed for a dovecote and in a private place - an exceptional leniency established because of the joy of Yom Tov; and taking the pigeons, which according to Beis Shammai requires moving them and taking hold of them on Erev Yom Tov, while according to Beis Hillel identifying them verbally is enough.