ASSALAM O ALAIKUM!
Saying “Ameen” after Surah Al-Fatiha is righteous (or Mustahab) in Islam. “Ameen” can be examplified with “Yaseen”.
It means “O Allah! Undertake my grants”.
Righteousness (Mustahab) of saying “Ameen” is backed by the Hadith of Wail Bin Hajar (r.a.) narrated by Masnad Ahmad, Abu Dawood & Trimidhi (r.a.) According to him, “I heard Muhammad (sal allah ho alaihi wasallim) saying “Ameen ” after reciting Surah Al Fatiha.
Our wise men say that one who is not in Salah should also say “Ameen” after the Imam recites Surah Al Fatiha. But one who is in Salah has more compulsion in saying “Ameen”.
Abu Hurera (r.a.) narrated that Muhammad (sal allah ho alaihi wasallim) said, “When your Imam says Ameen, you should also say the same. Whose ‘Ameen’ concides with Ameen of Angels, will be forgiven!” (SAHEHEEN)
This means that one a man & angels happen to say “Ameen” simultaneously, the sins of that man are forgiven by the Lord!
Hazrat Ibn e Abbas (r.a.) said to Muhammad (sal allah ho alaihi wasallim), “What is the meaning of Ameen?”
Muhammad (sal allah ho alaihi wasallim) said, “It means, O Lord, undertake my grant”
Trimidhi said, “It means, [O Allah] don’t break our hopes”
When one says Ameen over some prayer or Dua, it is as if he himself were praying to Allah, this is why when we say Ameen after the Imam recites Surah Al Fatiha, it is as we have also recited Surah Al Fatiha in Salah.
Let us make a habit of reciting Ameen whenever possible as it is one of the names of Allah.
Any mistake, on my part, if found, may please be forgiven.