The Reason Al-Fatihah is called Umm Al-Kitab
Al-Fatihah and the Prayer
Muslim recorded that Abu Hurayrah said that the Prophet(saw) said, "Whoever peforms any prayer in which he did not read Umm Al-Quran, then his prayer is incomplete." He said it thrice.
Abu Hurayrah was asked. "[When] we stand behind the Imam?" He said, "Read it to yourself, for I heard the Messenger of Allah(saw) say, "Allah, the Exalted, said, "I have divided the prayer (Al-Fatihah) into two halves between Myself and My servant, and My servant shall have what he asks for."
If he says,
Alhamdulillahi Rabbil Alameen (All praise and thanks be to Allah, the Lord of existence.)
Allah says, "My servant has praised Me." When the servant says,
Ar-Rahmanir Raheem (The Most Gracious, the Most Merciful.)
Allah says, "My servant has glorified Me." When he says,
Maliki Yawmid Deen (The Owner of the Day of Recompense.)
Allah says, "My servant has glorified Me, or My servant has related all matters to Me." When he says,
Iyaka Na'budu wa Iyaka Nastaeen (You (alone) we worship, and You (alone) we ask for help.)
Allah says, "This is between Me and My servant, and My servant shall acquire what he sought." When he says,
Ihdinas Siratal Mustaqeem, Siratal Ladheena An amta Alayhim, Ghayril Maghdoubi Alayhim Wa La dhalleen (Guide us to the straight path. The way of those on whom You have granted Your grace, not (the way) of those who earned Your anger, nor of those who went astray.)
Allah says, "This is for My servant, and My servant shall acquire what he asked for."
These are the words of An-Nasa'i, while both Muslim and An-Nasa'i collected the following wording, "A half of it is for Me and a half for My servant, and My servant shall acquire what he asked for."
Explaining this Hadith
The last Hadith used the word [Salah] "prayer" in reference to reciting the Quran, (Al-Fatihah in this case) just as Allah said in another Ayah,
[And offer your Salah (prayer) neither aloud nor in a low voice, but follow a way between.]
Meaning, with your recitation of the Quran, as the Sahih related from Ibn Abbas. Also, in the last Hadith, Allah said, "I Have divided the prayer between Myself and My servant. My two halves, a half for Me and a half for My servant. My servant shall have what he asked for." Allah next explained the division that involves reciting Al-Fatihah, demonstrating the importance of reciting the Quran during the prayer, which is one of the prayer's greatest pillars. Hence, the word prayer was used here although only a part of it was actually being referred to, that is, reciting the Quran. Similarly, the word "recite" was used where prayer is meant, as demonstrated by Allah's statement,
[And recite the Quran in the early dawn. Verily, the recitation of the Quran in the early dawn is ever witnessed.] in reference to the Fajr prayer. The Two Sahihs recorded that the angels of the night and the day attend this prayer.
Källa: Tafsir Ibn Kathir
The Tafsir of Surat Al-Fatihah
The Meaning of Al-Fatihah and its Various Names
Opener of the Book, also called, Umm Al-Kitab (the Mother of the Book)
Hadith recorded by At-Tirmidhi, who graded it Sahih, Abu Hurayrah said that the Messenger(saw) of Allah said, "Al-Hamdu lillahi Rabbil-Alamin is the Mother of the Quran, the Mother of the Book, and the seven repeated Ayat of the Glorious Quran."
It is also called Al-Hamd and As-Salah, because the Prophet(saw) said that his Lord said, "The prayer (Al-Fatihah) is divided into two halves between Me and My servants. When the servant says, All praise is due to Allah, the Lord of existence, Allah says, My servant has praised Me."
Al-Fatihah was called the Salah, because reciting it is a condition for the correctness of Salah. Al-Fatihah was also called Ash-Shifa(the cure). It is also called Ar-Ruqyah(remedy), since in the Sahih, there is the narration of Abu Said telling the story of the Companion who used Al-Fatihah as a remedy for the tribal who was poisoned. Later, the Messenger(saw) of Allah said to a Companion, "How did you know that it is Ruqyah?"
Al-Fatihah was revealed in Makkah as Ibn Abbas, Qatadah and Abu Al- Aliyah stated. Allah said,
[And indeed, We have bestowed upon you the seven Mathani](seven repeatedly recited verses), (Surat Al-Fatihah)(15:87) Allah knows Best.
Källa: Tafsir Ibn Kathir