Our Sages enacted that the Amida should be repeated by the Shaliah Tzibbur so that any congregant who was not proficient in praying the silent Amida could fulfill one’s obligation. The Tur writes that another reason that this was instituted was so that the congregation would have the opportunity to answer the Kedusha as well as Modim DeRabbanan. The Rosh (Megillah, 3:6) adds that it was enacted so that Birkat Kohanim could take place as well. As such, the Shulhan Aruch (Orah Haim 124:3) rules that, even if everyone is proficient in praying the silent Amida, the Shaliah Tzibbur must still repeat the Amida, so as to fulfill our Sages’ enactment.
The Rama of Panno writes (She’elot Utshuvot § 102:8) explains that if one did not have proper concentration in one’s silent Amida, then when one answers “Amen” intently to the blessings during the repetition it is as though one prayed with the proper concentration. The Arizal explains the repetition is on a higher level than the silent Amida since the silent Amida encompasses the lower Sefirot (Kabbalistic levels) of Netzah, Hod and Yesod while the repetition encompasses Hesed, Gevura and Tiferet. Furthermore, repeating the Amida out-loud has the power to prevent any negative forces from attaching themselves to our prayers.
Summary: There are several reasons for which the repetition of the Amida was instituted by our Sages.