Shaykh 3Uthaymeen may Allaah have mercy on him, said about fasting the six days of shawwaal:
“The wisdom behind fasting the six days of Shawwaal is the same as the wisdom regarding other optional acts which Allaah has ordained for His Servants in order to make the obligatory actions complete. So fasting the six days of Shawwaal is like the supererogatory Sunnah prayers [ar-Raatibah] which are performed after the obligatory prayers in order to complete any deficiencies in them.
And from Allaah, the Most High’s, Wisdom is that He made Sunnah [prayers] for the obligatory ones in order to complete them and raise them [in rank], so fasting the six days of Shawwaal has this great benefit.” (1)
The Prophet SallAllaahu 3Alayhi wa Sallam 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)
Shawwaal [page updated]References: (1) Fataawaa Noorun-3Alaa Darb. (2) Narrated by Abu Dawood (http://islamqa.info/en/ref/7859)