The meaning of Amen:[1]

The word Amen is an acronym for the words Keil Melech Neman, [which means that Hashem is a trustworthy king and will fulfill the blessing].  Amen is an expression of belief coming from the word Hamanas Devarim [i.e. Emuna].^[2]

The intent of the Amen:^[3] The intent that one is to have in mind upon answering Amen is that the blessing that the person said is true and I believe in this.^[4] This intent however is only applicable by those blessings that are factual statements of Hashem’s greatness [such as Baruch Sheamar, Yishtabach, and Goal Yisrael^[5]]. However, those statements that are a request and appeal towards Hashem, such as the stanzas of Kaddish [which is a request that Hashem return his Shechina] and the like, one is to intend on the future, that the words will come true and his requests will be speedily fulfilled.^[6] Those statements that are both admiration and praise and also hope and anticipation that the words being said will be speedily fulfilled, it is proper to also intend in the Amen both interpretations; that the statement said is true and that his words will come true and it will be speedily fulfilled.^[7] An example of such blessings are the middle blessings of Chazaras Hashatz, as well as the blessings of Ritzei and Sim Shalom.

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[1] Admur 124/11; 61/4; Levush 124/4

[2] Admur 124/9; Tur 124

[3] Admur 124/9

[4] Admur ibid; Michaber 124/6; Tur 124; Abudarham

[5] M”A 124/10

[6] Admur ibid; M”A ibid

[7] Admur ibid; Taz 124/3; M”A ibid