It is recommended(sunnah) for the Muslim to fast six days of Shawwaal as it has lots of virtues Prophet (PBUH) said: “Whoever fasts Ramadaan and follows it with six days of Shawwaal, it will be as if he fasted for a lifetime.”(1)
and reported in other hadith “as if he had fasted a whole year“
The Prophet (PBUH) explained: “Whoever fasts for six days after (Eid) al-Fitr has completed the year:” According to another report: “Allaah has made for each hasanah ten hasanat so 1=10, it means one month of fasting equal to ten months, and six days of fasting equal to tow months thus it completes the year.” (Alhumdulilah)
Another of the important benefits of fasting six days of Shawwaal is that is makes up for any shortfall in a person’s obligatory Ramadaan fasts, because no one is free of shortcomings or sins that have a negative effect on his fasting. On the Day of Resurrection, some of his naafil deeds will be taken to make up the shortcomings in his obligatory deeds, as the Prophet (peace and blessings of Allaah be upon him) said: “The first thing for which people will be brought to account on the Day of Resurrection will be their salaah (prayer). Our Lord, may He be glorified and exalted, will say to His angels – although He knows best – ‘Look at the salaah of My slave, whether it is complete or incomplete.’ If it is perfect, it will be recorded as perfect, and if something is lacking, He will say, ‘Look and see whether My slave did any voluntary (naafil) prayers.’ If he did some voluntary prayers, [Allaah] will say, Complete the obligatory actions of My slave from his voluntary actions.’ Then all his actions will be dealt with in a similar manner.” (2)
(1) (Narrated by Muslim, Abu Dawood, al-Tirmidhi, al-Nisaa’i and Ibn Maajah).
(2) (Narrated by Abu Dawood).