Why he started saying it so often
Aisha noticed the frequency before she knew the reason.
سُبْحَانَ اللَّهِ وَبِحَمْدِهِ أَسْتَغْفِرُ اللَّهَ وَأَتُوبُ إِلَيْهِ
Subḥāna-Llāhi wa bi-ḥamdih, astaghfiru-Llāha wa atūbu ilayh
“Glory to Allah and praise to Him. I ask Allah for forgiveness and I turn back to Him.”
The sign was the opening of Makkah. What it meant was that his work was finishing.
Where this comes from
She said: I see you saying this a great deal. He said his Lord had told him he would see a sign in his community, and that when he saw it he should say this often. And he said: I have seen it. Then he recited the surah about the help of Allah coming, and the people entering the religion in crowds.
The full narration
A'isha reported:The Messenger of Allah (ﷺ) recited often these words: Hallowed be Allah and with His praise, I seek the forgiveness of Allah and return to Him. She said: I asked: Messenger of Allah, I see that you often repeat the saying" subhan allahi bihamdihi astag firullahi watubuilaih" whereupon he said: My Lord informed me that I would soon see a sign in my Ummah, so when I see it I often recite (these) words: Hallowed be Allah and with His Praise, I seek forgiveness of Allah and return to Him. Indeed I saw it (when this verse) was revealed:" When Allah's help and victory came, it marked the victory of Mecca, and you see people entering into Allah's religion in troops, celebrate the praise of Thy Lord and ask His forgiveness. Surely He is ever returning to Mercy
Source
Ṣaḥīḥ Muslim · 1088
- Agreed upon as authentic (al-Bukhārī / Muslim)