“The emigrant (muhājir) may stay in Makkah for three (days) after completing his rituals (Hajj or 'Umrah).”

Narrated by al-Bukhārī (no. 3933) and Muslim (no. 1352, the wording is his), from the ḥadīth of al-‘Alā’ ibn al-Ḥadrami -may Allah be pleased with him-.
In another wording of both: "...after the Farewell Tawaf (aṣ-ṣadar)."

Brief Explanation of the Hadith

When the persecution of the Muslims in Makkah intensified, Allah –may He be exalted– permitted His Prophet –may Allah’s peace and blessings be upon him– to emigrate to Madīnah. The believers then emigrated, leaving their homes and wealth in pursuit of Allah’s pleasure, thereby attaining a great reward.
After Makkah was conquered and the emigrants arrived, the Prophet –may Allah’s peace and blessings be upon him– did not permit them to remain there for more than three days. He said: “The emigrant may stay in Makkah for three days after completing his ritual.” This means that one who emigrated before the conquest and then returned for Hajj, ʿUmrah, visitation, or trade should not stay longer than three days after completing his rituals, so that the reward of his emigration remains intact until the Day of Judgment.
In another narration it is stated: “Three days for the emigrant after aṣ-ṣadar,” meaning after the Farewell Tawāf. This is a concession allowing him to attend to his needs before returning to his place of emigration. It serves as a reminder that he left that town for the sake of Allah, and he should not remain there longer after departing in pursuit of Allah’s pleasure, so that the reward of emigration remains as long as his intention and devotion to Allah persist.
The stay was limited to three days because anyone who remained longer would be considered among the residents. Whoever stays in Makkah after having left it for the sake of Allah has effectively returned to what he abandoned for Allah. The principle is that whatever one abandons for the sake of Allah should not be reclaimed, in order to preserve the sincerity of one’s intention and maintain steadfastness in one’s emigration for Allah and His Messenger.