Fasting six days in Shawwal after the obligatory fast of Ramadan is Sunnah Mustahabbah. Although not obligatory, it is highly recommended for Muslims to do it. 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 according to a sahih hadith from the Prophet (pbuh). Abu Ayyub (may Allah be pleased with him) reported that the Messenger of Allah (pbuh) said:
“Whoever fasts Ramadan and follows it with six days of Shawwal, it will be as if he fasted for a lifetime.”
Thus, fasting in 6 days of Shawwal is a sign that one has increased in faith compared to Muslims who do not. It signifies that he is attaining the Love of the Most Merciful and becoming prominent in front of Him. In short, fasting in the six days after Eid ul-Fitr is an exceedingly blessed action that is not to be taken lightly.
May Allah accept (the fast and worships) from us, and from you. We hope you have a blessed Ramadan, and may Allah bless you with many more blessings!