It is our duty to praise the Master of All.
The Kaf HaChaim (132:11) quotes Shalmei Tzibbur, who explains the significance of Aleinu as serving as eidim, witnesses, attesting to the veracity and sincerity of one’s tefillah. Aleinu begins with an ayin and ends with a daled (ein od). Likewise, V’al kein nekave begins with an ayin (al) and concludes with a daled (u’Shemo echad). Twice, the word eid, witness, occurs; two witnesses who speak on behalf of the petitioner.
Rav Hai Gaon writes that one should recite Aleinu b’atifas ha’rosh, Tallis over one’s head, indicating his overwhelming respect and awe for the prayer which the Gaon says is the greatest praise of Hashem.
The Seder Ha’Yom cites a debate concerning the recitation of Aleinu following Minchah, since Shema Yisrael is not recited during this tefillah. He contends that, although we are not me’yacheid, declaring Hashem's Oneness through Shema Yisrael, the tefillah of Aleinu takes its place. This, in and of itself, demonstrates the significance and efficacy of Aleinu l’Shabeiach. The Radbaz (Teshuvos) posits that Aleinu following Minchah takes the place of Shema Yisrael. This is obvious from the recitation of Aleinu during the Mussaf prayers of Rosh Hashanah and Yom Kippur, despite the lack of Shema Yisrael in Mussaf.
