Abu Ayyoub (RA) narrated that the Messenger of Allah (SAW) said,
"He who fasts Ramadhan and follows it with six says of Shawwal, will be as if he had performed a perpetual fast." (Muslim).
Also, Thawban (RA) reported that the Prophet (SAW) said,
"Fasting Ramadhan is equal to fating ten months, and fasting six days of Shawwal is equal to two months. This is equal to fasting the whole year." (Ahmad & An-Nasa'i)
Therefore, fasting six days of Shawwal after having fasted Ramadhan will attain the person the reward of a perpetual fast, and more.
Furthermore, fasting Ramadhan, with belief and sincerity, erases all Previous sins, as the authentic Hadiths state. The ones who have performed the fast are given this bounty at the end of Ramadhan in Eid, which is the day of rewards. Therefore, fasting again after being given this great bounty serves as an act of thanking Allah, because no bounty is greater than having your sins forgiven. As for those who revert to their old evil ways after Ramadhan has passed, then they are indeed the losers. They loose Allah's bounty and they do not thank Him for anything.
Also, the good deeds that are performed in Ramadhan do not stop after the month ends, but they can still be performed as long as the person is alive and breathing. Fasting after Ramadhan is like the warrior who attacks and withdraws, only to attack again. This is unlike the coward who withdraws and runs away from the battlefield. Many people today are most happy when Ramadhan ends, because they see it as a burden on them and a task that one HAS TO DO and that THERE IS NO WAY OUT OF IT. Anyone with such a mentality and attitude will not be eager to fast again shortly after Ramadhan ends. On the other hand, the person who fasts again right after Ramadhan will be demonstrating his will and eagerness to fast for Allah's sake and his need to acquire Allah's pleasure.
Information from The Muslim Creed, November 2000 Vol. 8 No. 11