Because of this saying of Allah, exalted be His praise: "And remember Allah during a few numbered days." [Surah al-Baqarah: 203]
So Allah Almighty kept it general and did not restrict it, and He said: "Then when you have completed your rites of Hajj, remember Allah as you remember your forefathers, or with even greater remembrance." [Surah al-Baqarah: 200]
Hadith Referenceالسنن الكبرى للبيهقي / كتاب صلاة العيدين / Q6283
Abu Umayr, son of Anas bin Malik (may Allah be pleased with him), who was his eldest son, says: My paternal uncles from the Ansar, who were Companions of the Messenger (may Allah be pleased with them), said: The crescent of Shawwal was hidden from us, so in the morning we kept the fast. A caravan came at the end of the day, and they testified that they had seen the crescent the previous evening. He (peace and blessings be upon him) ordered the Companions to break the fast and said: "Go out for the Eid tomorrow morning."
This chain is sahih (authentic); the uncles of Abu Umayr from among the Companions of the Messenger of Allah (peace and blessings be upon him) can only be trustworthy.
Imam al-Shafi'i (may Allah have mercy on him) says: If this is established, that is right; if they do not go out for the Eid the next day, they may go out the day after, or else offer the Eid prayer on that same day after the sun passes its zenith.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب صلاة العيدين / 6283