Numerous laws are derived from the verse: "On the sixth day they shall prepare what they bring in..." (Shemot 16:5).
First, we derive from it that one should prepare properly on Friday for Shabbat, so that everything will be ready by the time Shabbat starts.
Second, we derive the rule of muktzah: if an item was not prepared or set aside for Shabbat use in advance, it may not be used or moved on Shabbat.
Third, our Sages derive from the verse that one may prepare on a weekday for Shabbat, but may not prepare on Shabbat for a weekday. For this reason, many people do not wash dishes or pots following Shabbat lunch, because they know they will not need to use them again until after Shabbat. Some people do not fold their tallit after shul, as they consider it preparing for a weekday since they will not be wearing a tallit again until Sunday.
Based on the requirement to prepare during the week for Shabbat, our Sages derive that if Yom Tov is on Friday, it is prohibited to prepare on Yom Tov for Shabbat. The only way this preparation becomes permitted is if a person sets aside food for an eruv tavshilin before Yom Tov. By doing so, he is beginning preparations for Shabbat on the day preceding Yom Tov.
Up to this point, we have addressed preparation undertaken by people. However, why do we need the verse cited above to tell us about such preparation? We have another verse which makes the same point: “Tomorrow is a day of rest . . . so bake what you want to bake now” (Shemot 16:23).
Therefore, the Gemara posits that our verse is speaking about something that was “prepared by heaven,” such as an egg that was laid on Shabbat. (This is one of the main subjects of the beginning of Tractate Beitzah). Such an egg may not be used on Shabbat or the Yom Tov that follows it on Sunday. Similarly, if Yom Tov is on Friday, an egg laid on Yom Tov may not be used for Yom Tov or the Shabbat following it. Since these eggs did not exist before Shabbat or Yom Tov, they could not have been prepared or set aside beforehand.