Chazal state in Tanna D'bei Eliyahu that "When the Yidden were in Mitzrayim, they all gathered and dwelt together because they were all together in one bond. They made a covenant to do loving kindness with each other, holding onto the covenant of Avraham, Yitzchak, and Yaakov, and serving their Father in Heaven alone."
These two covenants, being bonded as one, and serving only Hashem, are not separate from each other; they are interconnected. The Yidden's unity, "being in one bond", stemmed directly from their faith in one God.
The Midrash highlights a striking difference between the families of Eisav and Yaakov. When Eisav left Canaan, his family consisted of only six members, himself and his five sons, and the Torah refers to them in the plural, as "נפשות ביתו", the souls of his house. In contrast, when Yaakov's family came to Egypt, they were comprised of seventy members, yet they are described as a "נפש," a single soul. This is because they all served one God, and their shared faith in Him created a powerful sense of unity among them.
On this note, the Chasam Sofer ponders the concept of how תפילה can be effective when one davens for another. After all, Hashem is not like a human being who, when one person is unable to convince him, another person may be successful in doing so. How is it possible that the fate of a person can be altered through the prayer of another individual?
The Chasam Sofer concludes that this is because Yidden are considered to be part of one collective soul. This implies that one's suffering hurts the other. When one Jew prays for another, he says, it is as though the mouth, which is one limb of a single person, is praying for different parts of their own body. When so united, one Yid who davens for the other is essentially davening for himself.
The Gemara teaches that if the one in need of רחמי שמים is a Talmud Chacham, merely davening for him is not sufficient. Rather, one must make himself ill from worry for him. The Chasam Sofer says that this too indicates why one can daven for a different individual. For when a Yid shows that his fellow's suffering pains him as if it is his own, his tefillos are considered as an individual's prayer for himself.