“Fasting is on the day you all fast, breaking the fast is on the day you all break the fast, and the Day of Sacrifice is on the day you all offer the sacrifice.”

Narrated by al-Tirmidhī (no. 697), with the wording being his, and by Ibn Mājah (no. 1660), from the ḥadīth of Abū Hurayrah—may Allah be pleased with him.
Authenticated in Ṣaḥīḥ al-Jāmiʿ (no. 3869) and Silsilat al-Aḥādīth al-Ṣaḥīḥah (no. 224).

Brief Explanation of the Hadith

Among the objectives of the Islamic Sharīʿah is the unification of the Muslims and the rejection of division, especially in the outward rites in which unity and collective observance are manifest. Among this is his statement—may Allah’s peace and blessings be upon him— “Fasting is on the day you all fast, and breaking the fast is on the day you all break the fast.” Its meaning is that the fasting of Ramaḍān and the breaking of the fast are legally considered according to what the Muslim community and their leader observe, and “the Day of Sacrifice is on the day you all offer the sacrifice,” meaning that the commencement of ʿĪd al-Aḍḥā is determined in accordance with what accords with the Muslim community and their leader.
If it were to occur that an error happened in the sighting of the crescent, and the people fasted or broke the fast accordingly, then later it became clear otherwise, their act of worship remains valid and they bear no sin—except in the case that the month was not completed, in which case they must make it up. Likewise, with ʿĪd al-Aḍḥā: if they err in the sighting of the crescent, their standing is valid, their sacrifices are valid, and their Hajj is valid, by the will of Allah.
The apparent meaning of the ḥadīth is that one who alone sights the crescent is required to follow the Muslim community and their leader, in order to preserve unity and block the means to division. This is more appropriate than acting upon individual judgment, and in this matter reliance is not placed upon astronomical calculations, but rather upon that which the Ummah agrees upon.