"Ein motzi'in es ha'esh":
The Mishnah rules that one may not bring forth fire on Yom Tov, meaning that one may not kindle a new fire. The Mishnah lists four methods:
Min ha'eitzim - by rubbing sticks against one another.
Min ha'avanim - by striking stones against one another until a spark is produced.
Min he'afar - a hard earth that had the capacity to ignite a fire, possibly because it contained certain chemical substances.
Min hamayim - they would place water inside a glass, something like a magnifying glass, and by means of the sun's rays they would ignite something with it.
The reason for the prohibition is molid: since it appears as though one is generating something new, it looks like a melachah, and the Sages forbade it. One is permitted only to take fire from an already existing fire.
"V'ein masikin es hare'afim litzlos bahen":
The Gemara explains that the Mishnah is speaking of new roof tiles. Heating and whitening them while they are new - whitening meaning bringing them to a glowing, white heat - constitutes completing a vessel, since this is the final stage of their manufacture. For this reason one may not heat them on Yom Tov, since by doing so he completes them.
Rabbi Eliezer's leniency in muktzeh:
Rabbi Eliezer, whose opinion already appeared in the previous Mishnah regarding taking a splinter to pick one's teeth, is mentioned in our Mishnah as well with a further leniency - a leniency concerning muktzeh. It should be emphasized that muktzeh here is not the familiar concept of an item that a person has mentally set aside and that may not be handled on Yom Tov, but rather the name of a place: the area where figs and grapes were spread out to dry.
According to Rabbi Eliezer, a person stands beside the muktzeh on Erev Shabbos during the Shemittah year and says: "Mikan umikan ani ochel lemachar" - from this area and from that area I will take to eat tomorrow, when the fruits will be dry and fit to eat. In his view, preparation through such a general statement is sufficient.
Why did the Mishnah specify Erev Shabbos of the Shemittah year? Because in other years, fruits lying in the muktzeh to dry have not yet had Terumah and maasros separated from them, and therefore they are not ready to be eaten, and one cannot separate Terumah and maasros from them on Shabbos or Yom Tov. The Mishnah therefore speaks of the Shemittah year, in which there is no obligation to separate Terumah and maasros, so that the only remaining problem is the need to identify them in advance, before Shabbos begins, as fruits that he intends to eat.
And the reason this preparation is needed: once a person placed them in the muktzeh before Shabbos so that they would dry, he thereby set them aside from being eaten. He must therefore nullify that setting aside and identify them as fruits that he intends to eat, and then they are no longer muktzeh.
The Sages disagree and say: "Ad sheyirshom veyomar: mikan v'ad kan" - until he actually identifies the fruits that he intends to eat. A general statement, as Rabbi Eliezer ruled, that somewhere in this muktzeh he will eat tomorrow, is not enough.
And some say that the point of dispute between Rabbi Eliezer and the Sages is whether we say 'yesh bereirah': can we say that the fruit he ultimately takes is retroactively clarified to be the one he intended to eat from the outset. The Sages hold that the matter is not clarified retroactively; rather, one must identify before Shabbos which fruits are no longer muktzeh and are designated for eating.