The Gemara brings a story with Rebi Meir: there were people in his neighborhood who would aggravate him a lot, so he davened that they should die. His wife, Beruria heard his davening and responded, “It statesיתמו חטאים , the sins should be finished off, however it does not say חוטאים, the sinners should be finished off. Alos, the passuk ends ורשעים עוד אינם and they won’t be wicked anymore. Therefore, don’t daven that they should die, but rather daven that they should do teshuva.” Rebi Meir listened to his wife, and davened for them and they indeed did teshuva.
The Maharsha asks: it is understood that if a person davens for himself to do teshuva and fix his ways, Hashem will help him to return because Chazal say, “In the way that a person wants to go, Hashem will lead him, and one who wants to purify himself, Hashem will purify him.” Yet, Chazal say, “הכל בידי שמים חוץ מיראת שמים, everything is from Hashem besides for the fear of Hashem.” How then can one daven for someone else to do teshuva?
HaRav Moshe Feinstein zatzal answers that Rebi Meir davened that they should not have challenges in order to make it easier for them to do teshuva. Also, one can daven for someone that Hashem should help him find the tzaddikim and influential people to guide him to the truth.
HaRav Chaim Kanievsky shlita answers b’shem the Chazon Ish: Hashem gives a person freewill and doesn’t automatically put into one’s heart the feeling of closeness to Hashem to be aware of sin. Yet one Yid can cause another person to serve Hashem by convincing him or making him do what is right. This is because Klal Yisrael is one (ערבים זה לזה) and every action by one person can influence and inspire others. Therefore, if a person davens for another person to do teshuva, Hashem accepts the tefillah and sends messages to the person to be alert not to sin. So too, Rebi Meir davened that they should do teshuva, and Hashem caused them to be influenced and inspired to return to Hashem.
On Yom Kippur we daven אבינו מלכנו חטאנו לפניך, “Our Father our King, we sinned before You…,” which is in the plural form, and not חטאתי, I sinned, in the singular form. According to Harav Moshe Feinstein we are davening for others to have less challenges and to find the right people to influence and inspire them. Rebi Umori HaRav Yaakov Friedman shlita explained with the Chazon Ish that Klal Yisrael is one, and my davening will indeed cause Hashem to send inspiration and influences to turn others upon the proper path.
May Hashem answer our davening while the entrance is still open, to bring all of Klal Yisrael close to Hashem!