How did Aisha (may Allah be pleased with her) hear the Prophet's (peace be upon him) supplication while prostrating: "O Allah, I seek refuge in Your pleasure from Your wrath..." even though the basic principle of supplication is to be private?

The remembrances (adhkar) during bowing (ruku') and prostration (sujud) are, by consensus, to be recited silently. "Silently" (israr) means that the worshipper hears himself but no one else. However, this does not prevent reciting them aloud (jahr) occasionally for a need, such as teaching or intimate supplication (munajat). The Prophet, peace and blessings be upon him, sometimes recited a verse aloud in a silent prayer to demonstrate its permissibility. The limit of audible recitation (jahr) is for those nearby to hear, and silent recitation (israr) is for the worshipper to hear himself. An intermediate level between these two is permissible. The Prophet, peace and blessings be upon him, once recited the supplication during prostration aloud, to the extent that Aisha, may Allah be pleased with her, heard him. This slight audibility is understood as stemming from the intensity of his intimate supplication. Reciting aloud for a temporary reason is good, whether for teaching or another benefit, but it should not be done consistently. Some Companions also recited some remembrances aloud in prayer, and the Prophet, peace and blessings be upon him, praised whoever recited a remembrance aloud after the Takbir. Silent recitation (israr) requires the movement of the tongue, and someone very close might hear it. Aisha might have heard the Prophet, peace and blessings be upon him, after she drew very close to him in the dark.