Just a note - from what I can see, this hadith in this form may only be from shia sources. The sunni source of Sahih Muslim has:

'A'isha reported that Allah's Apostle (may peace be upon him) went out one norning wearing a striped cloak of the black camel's hair that there came Hasan b. 'Ali. He wrapped hitn under it, then came Husain and he wrapped him under it along with the other one (Hasan). Then came Fatima and he took her under it, then came 'Ali and he also took him under it and then said: Allah only desires to take away any uncleanliness from you, O people of the household, and purify you (thorough purifying)

and after that there is a case of context from Shaykh Wikipedia which is also interesting to read all the ways in which this hadith is taken.

It seems from the wikipedia link that the shia use this hadith to support their idea of the imamate etc.

PS I am not suggesting that the above version is also not in sunni sources, or that it is wrong - but just that its not totally clear cut.