What are the main reasons we are in exile?
I recall some reasons but I can't remember all of them.
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The Gemara Pesachim 87B:
וְאָמַר רַבִּי אֶלְעָזָר: לֹא הִגְלָה הַקָּדוֹשׁ בָּרוּךְ הוּא אֶת יִשְׂרָאֵל לְבֵין הָאוּמּוֹת אֶלָּא כְּדֵי שֶׁיִּתּוֹסְפוּ עֲלֵיהֶם גֵּרִים, שֶׁנֶּאֱמַר: ״וּזְרַעְתִּיהָ לִי בָּאָרֶץ״. כְּלוּם אָדָם זוֹרֵעַ סְאָה אֶלָּא לְהַכְנִיס כַּמָּה כּוֹרִין.
And Rabbi Elazar said: The Holy One, Blessed be He, exiled Israel among the nations only so that converts would join them, as it is stated: “And I will sow her to Me in the land” (Hosea 2:25). Does a person sow a se’a of grain for any reason other than to bring in several kor of grain during the harvest? So too, the exile is to enable converts from the nations to join the Jewish people
On page 12 of Rabbi Menachem Mendel Schneerson's (the Lubavitcher Rebbe) book "Torah Studies a parsha anthology", he cites a Midrash (Bereishit Rabbah 39:2) about Abraham's journeys and writes,
"The Midrash compares his many journeyings to the way a spice box must be shaken about to spread its aroma to all corners of a room."
A few pages later (p.14) the Rebbe writes about exile saying,
"Exile, then, is an integral part of spiritual progress; it allows us to sanctify the whole world by our actions, and not simply a small corner of it."
(see @Shmuel's link in the comments for an on-line version of the Rebbe's words)
In my mind, when I read these passages I like to think that Israel is the spice box and by scattering Jews (exile) into the four corners of the world we bring G-d's light to the darkest fringes so that others may see it and be guided by it, and thereby hasten the appearance of the Messiah.
I mean the Torah clearly illustrates the connection between the Jews and the land of Israel- when we sin and don’t follow in the ways of G-d, the land will kick us out. So galus is our punishment for not following the Torah, and it gives us a chance to repent and think about what we’ve done and better ourselves
והוצאתי אתכם מן העמים וקבצתי אתכם מן הארצות אשר נפוצותם בם ביד חזקה ובזרוע נטויה ובחמה שפוכה. והבאתי אתכם אל מדבר העמים ונשפטתי אתכם שם פנים אל פנים. כאשר נשפטתי את אבותיכם במדבר ארץ מצרים כן אשפט אתכם נאום אדני ה'. והעברתי אתכם תחת השבט והבאתי אתכם במסרת הברית. וברותי מכם המרדים והפושעים בי מארץ מגוריהם אוציא אותם ואל אדמת ישראל לא יבוא וידעתם כי אני ה
And I will bring you out from the peoples, and will gather you out of the countries wherein ye are scattered, with a mighty hand, and with an outstretched arm, and with fury poured out; and I will bring you into the wilderness of the peoples, and there will I plead with you face to face. Like as I pleaded with your fathers in the wilderness of the land of Egypt, so will I plead with you, saith the Lord GOD. And I will cause you to pass under the rod, and I will bring you into the bond of the covenant; and I will purge out from among you the rebels, and them that transgress against Me; I will bring them forth out of the land where they sojourn, but they shall not enter into the land of Israel; and ye shall know that I am the LORD.