Praise be to Allaah.
Fasting six days of Shawwaal after the obligatory fast of Ramadaan is Sunnah Mustahabbah, not waajib. It is recommended for the Muslim to fast six days of Shawwaal, and in this there is great virtue and an immense reward. Whoever fasts these six days will have recorded for him a reward as if he had fasted a whole year, as was reported in a saheeh hadeeth from the Prophet (peace and blessings of Allaah be upon him). Abu Ayyoob (may Allaah be pleased with him) reported that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: Whoever fasts Ramadaan and follows it with six days of Shawwaal, it will be as if he fasted for a lifetime. (Narrated by Muslim, Abu Dawood, al-Tirmidhi, al-Nisaa'i and Ibn Maajah).
The Prophet (peace and blessings of Allaah be upon him) explained this when he said: Whoever fasts for six days after (Eid) al-Fitr has completed the year: (whoever does a good deed (hasanah) will have ten hasanah like it). According to another report: Allaah has made for each hasanah ten like it, so a month is like fasting ten months, and fasting six days completes the year. (al-Nisaa'i and Ibn Maajah. See also Saheeh al-Targheeb wa'l-Tarheeb, 1/421). It was also narrated by Ibn Khuzaymah with the wording: Fasting for the month of Ramadaan brings the reward of ten like it, and fasting for six days brings the reward of two months, and that is the fasting of the whole year.
The Hanbali and Shaafa'i fuqaha' explained that fasting six days of Shawwaal after fasting Ramadaan makes it as if one has fasted for an entire year of obligatory fasts, because the multiplication of the reward applies even to naafil fasts, because each hasanah brings the reward of ten like it.
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. (Narrated by Abu Dawood).
And Allaah knows best.