This hadeeth was narrated from the Mother of the Believers, ‘Aishah (may Allah be pleased with her) who said: 

The Messenger of Allah (peace and blessings of Allah be upon him) used to fast three days every month, and sometimes he would delay that until a year’s fasts accumulated and sometimes he would delay it until he fasted Sha‘ban. 

This was narrated by at-Tabarani in al-Mu‘jam al-Awsat, 2/320. He said: Ahmad told us: ‘Ali ibn Harb al-Jandeesaboori told me: Sulayman ibn Abi Hawthah told us: ‘Amr ibn Abi Qays told us: from Muhammad ibn ‘Abd ar-Rahman ibn Abi Layla, from his brother ‘Eesa, from his father ‘Abd ar-Rahman, from ‘Aishah (may Allah be pleased with her) who said. Then he quoted it, then he said: This hadeeth was not narrated from ‘Abd ar-Rahman ibn Abi Layla except via this isnad (chain of narrators), and ‘Amr is the only one who narrated it. End quote. 

This is a da‘eef (weak) isnad because of Muhammad ibn ‘Abd ar-Rahman ibn Abi Layla – the famous faqeeh (jurist) – of whom Imam Ahmad said: He had a poor memory and his narration of hadeeth was inconsistent. Shu‘bah said: I have never seen anyone with a poorer memory than Ibn Abi Layla. ‘Ali ibn al-Madeeni said: He had a poor memory and his hadeeth is weak. Hence the scholars classed this hadeeth of his as da‘eef. 

Al-Haythami (may Allah have mercy on him) said: 

Its isnad includes Muhammad ibn Abi Layla and there are some reservations concerning him. End quote. 

Majma‘ az-Zawaid, 3/195 

Al-Hafiz Ibn Hajar (may Allah have mercy on him) said: 

Ibn Abi Layla is da‘eef. The hadeeth of the door and the one that follows it are indicative of the weakness of what he narrated. End quote. 

Fath al-Bari, 4/252 

Ash-Shawkani (may Allah have mercy on him) said: 

Its isnad includes Ibn Abi Layla, who is da‘eef. End quote. 

Nayl al-Awtar, 4/332 

The scholars differed concerning the reason why the Prophet (peace and blessings of Allah be upon him) fasted most of Sha‘ban and there are many opinions, including the one mentioned above, but the evidence for it is not sound. Perhaps the first one who narrated it was Ibn Battal in his commentary on Saheeh al-Bukhari (4/115). He also mentioned other opinions that were narrated by al-Hafiz Ibn Hajar (may Allah have mercy on him) and added to them, then he said: 

More appropriate than this is what is mentioned in a hadeeth that is sounder than what has been mentioned above. It was narrated by an-Nasai and Abu Dawood, and classed as saheeh (authentic) by Ibn Khuzaymah, from Usamah ibn Zayd (may Allah be pleased with him) who said: I said: O Messenger of Allah, I have not seen you fasting in any month more than you do in Sha‘ban. He said: “That is a month that people are heedless of, between Rajab and Ramadan, but it is a month in which deeds are taken up to the Lord of the Worlds, and I like my deeds to be taken up when I am fasting.”

End quote from Fath al-Bari, 4/215 

And Allah knows best.