When saying “Amen” after hearing G‑d’s praise, it is taught that one’s intention should be that “the blessing that was recited is true and I believe in it,” since the word Amen signifies an affirmation of belief. The letters of the word אמן amen are the root letters of the word אמונה emunah, meaning belief or trust. That can be seen as well by the famous teaching that אמן is short for אל מלך נאמן. The Maharal explains that this is a simple praise for Hashem stating that He is faithful in paying the reward of those who serve Him.
Our sages teach that in some ways, the person who responds “Amen” is even greater than the one who said the blessing. When one responds “Amen” with the proper concentration it has the power to open the gates of Gan Eden and nullify negative decrees. (Talmud Shabbat 119b) The Leshem expands further on this and explains that by saying a blessing, one only affirms the truth of Hashem’s blessing to the world, but the word Amen both affirms the truth and expresses the wish for it to continue. That is why it is greater!
In a more mystical approach, The Rokeach teaches us that the angels created by the amens of Esther stayed with her in the palace of Achashveirosh, and protected her throughout the Purim story. Even if one chooses not to take that teaching literally and proclaims that no angels were created, it can be suggested that the amen itself kept her strong through all her struggles. We mentioned earlier that saying אמן amen instills אמונה (trust in Hashem) in a child. So perhaps did the Amens of Esther’s youth keep her connected to Hashem.
However, The Arizal, one of the most influential Kabbalists of all time states that when one says “Amen” with the proper concentration, there are indeed angels that are created who then respond “Amen” in unison.