In our Gemara on Amud Beis, there is a phrase quoted that is used almost exclusively by Rabbi Yehuda Hanasi, the Compiler of the Mishna. Rabbi Rosner's recorded Daf Yomi Shiur on this daf from the previous cycle brought attention to this unique usage (אומר אני). Rabbi Yehuda Hanasi introduces his halakhic argument with the phrase "I say." Unlike other Tannaim and Amoraim, he uniquely employs this expression (see Meleches Shlomo, Mishna Arakhin 4:2, for a list of its occurrences).
What does this expression signify? In English, it may carry a tone of arrogance, but that cannot be the intended connotation by Rabbi Yehuda Hanasi, who is known for his humility (see Mishna Sotah 9:15).
Rav Yosef Engel (Bais Haotzar, Klal 33) suggests that Rabbi Yehuda Hanasi used this phrase to present his perspective humbly, as if saying, "This is The Way, as I see it," acknowledging the possibility of other opinions as well.
Based on the teachings of the Baal Shem Tov, Rav Engel also proposes that the phrase implies speaking with divine guidance. Rabbi Yehuda Hanasi might have been saying, "I observe myself, saying," as if amazed by the words coming from his mouth, attributing his wisdom to divine inspiration rather than his own intellectual prowess.
Rabbi Reuven Margolis (Yesod Hamishna Va-Arichasa, 17-19) offers a simpler explanation. As the compiler of the Mishna, Rabbi Yehuda Hanasi often quoted and anthologized other teachings. When expressing something that was his own opinion or tradition, he would preface it with אומר אני, "I say," to avoid misleading others into thinking he was quoting a different opinion.
In line with Rav Engel's first explanation, this expression serves as a valuable communication tool to convey that you are speaking from your own perspective, which is more collaborative and encourages others to share their perspectives.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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