The context of this begins with the opening case of the Mishna where an egg is born on Yom Tov on which Beis Shammai say that it is permitted to be eaten on this Yom Tov day as there is no problem with the egg being Muktzeh, while Beis Hillel say that it is forbidden to be eaten on this Yom Tov day because the egg is considered Muktzeh (Note: In most cases, Beis Shammai is the strict view while Beis Hillel is the lenient view, but this is one of the few exceptions). On this, Rav Nachman states that the case of the Mishna is dealing with a hen that is set aside particular for producing eggs - as opposed to a hen set aside for eating which would otherwise not be an issue as being permitted to be schechted and eaten on Yom Tov. As Rav Nachman explains the difference of views, the ones who hold strictly about the laws of Muktzeh consider this to be an issue of Nolad and hence forbidden to be eaten on Yom Tov, while the ones who are lenient about the laws of Muktzeh don't consider this to be an issue of Nolad and hence permitted to be eaten on Yom Tov. With this said, Beis Shammai rule like Rabbi Shimon who follows the latter view while Beis Hillel rule like Rabbi Yehuda who follows the former view.
Now, the Gemoro questions that Rav Nachman said this, because in another context where Beis Shammai and Beis Hillel are also lenient or strict, respectively, pertaining to the laws of Shabbos, Rav Nachman is found to be saying that the views need to be switched so that Beis Shammai rule like Rabbi Yehuda who is the strict one as per Muktzeh laws while Beis Hillel rule like Rabbi Shimon who is the lenient one as per Muktzeh laws.
As it turns out, the reason for the difference between whom Beis Shammai and Beis Hillel hold depending in what case according to Rav Nachman as to do with whether it pertains to Shabbos or Yom Tov. When it comes to Shabbos on which everyone knows to be strict as per Muktzeh, the typical Mishna as per the laws of Shabbos is worded anonymously according to the lenient view of Rabbi Shimon, thus establishing that Beis Shammai rules like Rabbi Yehuda while Beis Hillel rules like Rabbi Shimon as per the lenient view. However, when it comes to Yom Tov on which some rules as per Melacha are relaxed particularly as per cooking (Meleches Ochel Nefesh) and hence there is a tendency for some to treat Yom Tov lightly as per Melacha, the typical Mishna as per the laws of Yom Tov is worded anonymously according to the strict view of Rabbi Yehuda, thus establishing that Beis Shammai rules like Rabbi Shimon according to the lenient view while Beis Hillel rules like Rabbi Yehuda.