“Whoever fasts six days after (`Eid) al-Fitr completes thereby (fasting) the year, for whoever brings a good deed will have it multiplied tenfold.”

Narrated by Aḥmad (no. 22412), al-Nasā’ī in al-Sunan al-Kubrā (no. 2874), and Ibn Mājah (no. 1715, this is his wording), from the ḥadīth of Thawbān — may Allah be pleased with him.
The wording of al-Nasā’ī states: “Allah has made each good deed worth tenfold; thus a month (of fasting) equals ten months, and (fasting) six days after al-Fiṭr completes the (reward of) a full year.”
Authenticated in Ṣaḥīḥ al-Jāmiʿ (no. 6328) and Ṣaḥīḥ al-Targhīb wa l-Tarhīb (no. 1007).

Brief Explanation of the Hadith

It was from the guidance of the Prophet — may Allah’s peace and blessings be upon him — to remain constant in acts of obedience. Whenever he completed one form of worship, he would follow it with another. Among these acts was his perseverance in the worship of fasting, which is one of the most sublime acts of worship and greatest in reward. He had voluntary fasts that he would observe every week and month, and he would encourage his companions to do the same.
In this ḥadīth, the Prophet — may Allah’s peace and blessings be upon him — guides us to the great virtue of fasting six days from the month of Shawwāl after Ramadān, due to the immense reward they carry. Fasting these six days is counted as if one had completed fasting the entire year, for each good deed is rewarded tenfold. Thus, the fasting of Ramadān equals ten months, and the six days equal two months, making a total of twelve months — as indicated by the Almighty’s saying: {Whoever comes on the Day of Judgment with a good deed will have ten times the like thereof.} (al-An‘ām 6:160).
This is among the manifestations of Allah Almighty’s grace and generosity toward this Ummah, for He has made deeds few in effort yet multiplied greatly in reward. A Muslim should therefore be keen to fast these six days—whether consecutively or separately, at the beginning, middle, or end of the month. However, hastening to fast them is preferable, in accordance with Allah’s saying: {So race to all that is good.} (al-Baqarah 2:148), for the servant does not know what circumstances may arise to prevent him.
Whoever still owes days from Ramadān, it is preferable for him to make up those missed fasts first and then follow them with the six days of Shawwāl. However, if that is not feasible, he may fast the six days first and make up the missed days later, though prioritizing the making up of missed fasts (qadā') is more appropriate and cautious.