Fasting six days in the month of Shawwal is mentioned in a hadith of the Prophet (peace and blessings be upon him). He is reported to have said, “Whosoever observes fast in Ramadan and then follows it up with six days from Shawwal, will be regarded as though he had been fasting every day.” (Reported by Muslim, Abu Dawud and at-Tirmidhi).
This is an authentic hadith which indicates that fasting six days of Shawwal is a Prophetic sunnah which the Imam Ash-Shafi`i, Ahmad, and a number of leading scholars and jurists followed.
Hence, it is wrong to oppose this authentic hadith simply because a scholar viewed it as makruh out of the fear that an ignorant person might think that it was part of Ramadan, or that some people might think it is an obligatory fast, or on the ground that he was not aware of an evidence showing that one of the earlier scholars used to fast these days. These are all speculations, thus, they cannot be used to oppose the authentic hadith mentioned above.
Based on this explanation, it becomes clear that fasting the six days of the month of Shawwal is a confirmed Prophetic Sunnah and not an innovation or a makruh act as some claim.