Why do we exist in the first place? No matter what God’s reason for creating us, why did He create us in such a way that we have to suffer? Why do humans have to try to exist together?
The first thing to acknowledge is that the Infinite (God) has a will, and the inner essence of His will is to reveal His transcendence of paradox and omnipotence allowing Him to indwell His infinite nature within finite reality, creating what is known as "the truly limitless within the limited" (the rabbinic sages called this, "a dwelling place below"). To this end, God created man who is a finite creation (i.e., man is neither limitless nor omnipotent).
Being Finite and Suffering
It is because man is finite that he experiences suffering (in Hebrew, the verbs for "suffering" [סבל] and for "finite" [גבל] are phonetically similar). The suffering experienced has two main sources, but in both cases, the suffering is the result of an attempt to expand these limits in an improper and un-Godly manner.
The first source of suffering in life stems from the individual's drive to expand his ego without limits. A rectified and God-fearing individual possesses true humility and does not feel the need to nurture and inflate his or her "sense of self." Thus, the rectified individual does not experience the barriers imposed by being a finite being, for they are really only barriers against the expansion of one's pride in one's self, one's feeling of importance, etc. The rectified individual only basks and takes pleasure in the light [i.e., the revelation] of God and has no experience of suffering. In addition, an individual who is humble will respect the limits and boundaries of others. In this vein, the Torah commands us, "Do not compromise your fellow's boundaries" (Deuteronomy 19:14), which literally refers to the boundaries of the property owned by your fellow, but also refers to challenging his spiritual and psychological limits.
These personal, individual barriers also include one's body, which according to Ezekiel's prophecy is composed of four elements: bones, sinews, flesh, and skin (see Ezekiel 37:1-14). In his prophecy, Ezekiel saw these four elements coming together and forming a human body and then saw how the breath of life entered this corpse and gave it life. The four elements correspond to the four letters of God's essential Name, Havayah (Yud, Hei, Vav, Hei), in order, and the breath of life that entered corresponds to the "tip of the Yud," or the crown. From this prophecy, it follows that the final barrier or limit in the human body is the skin, which is the organ that is afflicted by the one disease described in length in the Torah–the disease known as tzara'at (see Leviticus ch. 13 and following). The Hebrew word for "affliction" (נֶגַע) is a permutation of the word for "pleasure" (עֹנֶג), and thus, it is in the barrier of the skin that the pleasure from God's revelation that should have been his lot is transformed into an affliction stemming from the over-expansion of the ego.
The word for "finite" (גְּבוּל), which we have been discussing, also means the border or boundary of a state or country, which leads us to the second source of tremendous suffering in our world. It is not only the single individual who tries to expand his "sense of self-importance" and "self-worth" ultimately causing himself suffering. When the same thing happens to a king or other leader, the results can bring about pain and suffering not just for a single individual, but for an entire nation. Like the skin, which acts as the border between the individual's body and his surroundings (acting to keep out unwanted germs, etc.), a nation's borders, including both physical boundaries with other nations as well as the limits of a nation's influence on its neighbors and people that could be half a world away, can be positive. However, the quest to increase their national "honor" or "worth," has been the root of all the conflicts and wars between nations, causing them and others so much of the grief and suffering felt in this world.
The Importance of Society
During the course of our being in this world, which is the world that is the lowest of all possible worlds, we are entrusted with a great number of tasks for the sake of revealing the infinite in the finite, and the limitless in the limited. Many of these tasks cannot be completed alone and require the joint efforts of a number of individuals in order to reach a successful conclusion. This is similar to the Torah's description of the stone that covered the well in Haran (see Genesis 29:2-10). Moving the stone in order to draw the water required the cooperation of many shepherds who came together to water their flocks. Indeed, from time to time, there can be a great tzaddik–a pious individual–who represents a particular facet of the Mashiach, the redeemer, who can display supernatural powers (i.e., he channels the infinite through him) and can move the stone by himself; all of this is explained in length in Chasidic works, the teachings of the Ba'al Shem Tov.
This need to cooperate and work together is why man was created with the purpose of being "political," with the ability to join forces with others to seek the common good. The primary demonstration of "the infinite dwelling in the finite," or the task of making God a dwelling place here below, is achieved when we all (all of humanity) merit a pious and righteous king (the Mashiach) who will unite as all so that we can bring the limitless into the limited–the express goal of all the commandments of our holy Torah.