Hadith: The Messenger of Allah (may Allah's peace and blessings be upon him) went out to invoke Allah for rain. He faced the direction of Qiblah as he was invoking Allah, and turned his upper garment inside out. Then he prayed two Rak‘ahs and recited the Qur'an out loud in both of them

عن عبد الله بن زيد بن عَاصِم المازِنِي رضي الله عنه قال: «خرج النبي صلى الله عليه وسلم يَسْتَسْقِي، فتَوَجَّه إلى القبلة يدْعو، وحَوَّل رِدَاءه، ثم صلَّى ركعتين، جَهَرَ فيهما بالقِراءة». وفي لفظ «إلى الْمُصَلَّى».
[صحيح] - [متفق عليه]

‘Abdullāh ibn Zayd ibn ‘Āsim al-Māzini (may Allah be pleased with him) reported: The Messenger of Allah (may Allah's peace and blessings be upon him) went out to invoke Allah for rain. He faced the direction of Qiblah as he invoked Allah, and turned his upper garment inside out. Then he prayed two Rak‘ahs and recited the Qur'an out loud in them. In another wording: "...facing the place of praying."
[Authentic hadith] - [Narrated by Bukhari & Muslim]

Explanation

Allah, the Exalted, tests His slaves with many types of trials, so that they will invoke Him alone and remember Him. So, when the earth became barren during the lifetime of the Prophet (may Allah's peace and blessings be upon him) he went out with the people toward the place where Eid prayer is performed in the desert to ask for rain from Allah, the Exalted, and in order to show more humbleness and need to Allah, the Exalted. So, he faced the direction of Qiblah, where supplication is more likely to be answered, and began to invoke Allah to send down rain for the Muslims and to remove the drought that afflicted them. Out of optimism that their situation would change from drought and hardship to fertility and life of ease, he turned his upper garment inside out, then led the people in Istisqā' prayer. The prayer consisted of two Rak‘ahs in which his recitation of the Qur'an was out loud, because it was a largely congregational prayer.