The gemara says that Rebbi and R' Nasan were the end of the mishna era, and Rav Ashi and Ravina were the end of "hora'ah". Rashi explains that Rebbi was responsible for organizing the mishna but this is not to say that the mishna was closed so that no additions can be made after Rebbi. Rashi clearly acknowledges that ואחריהם לא יוסיפו אלא מעט - implying that there were bits and pieces added later. Similarly, when the gemara says that R' Ashi and Ravina were the end of the "hora'ah" period, rashi doesn't say that they had the final word and no later authority can argue on them. Rather, rashi says that they too were involved in organizing the Talmud as we know it, making sure the statements that were made fall under the mishna on which they belong. Rashi doesn't even mention the idea that no one can argue on the conclusions of Ravina and R' Ashi.
The one who addresses the issue is the Kesef Mishna (mamrim 2:1). After the Rambam writes that when a Beis Din HaGadol interprets a din, a later beis din has the authority to disagree and darshen the way they see it (which is not the case by a gezeira of an earlies beis din). Based on this the kesef mishna asks, why is it that amora'im don't argue on tana'im, and why is it that the generation after the amora'im don't argue on them? The kesef mishna answers:
ואפשר לומר שביום חתימת המשנה קיימו וקבלו שדורות האחרונים לא יחלוקו על הראשונים, וכן עשו גם בחתימת התלמוד שמיום שנחתם לא ניתן להם רשות לשום אדם לחלוק עליהם
He explains that the generation following the close of the Talmud had accepted upon themselves not to argue. The kesef mishna doesn't mention our gemara רב אשי ורבינא סוף הוראה, because he considers it irrllevant to the issue - it doesn't explain why the later generation can't argue. Therefore, the kesef mishna makes an assumption that it was just accepted that no one would argue on the gemara after it was sealed.
The question still remains, what makes this binding? What if someone would argue on the gemara - we would surely call him a heretic, but why?
R' Elchonon Wasserman (kuntros divrei sofrim 2:6) explains that the closing of the mishna was done with a gathering of all chachmei yisroel who have the power of a beis din hagadol to prevent anyone from arguing. He argues that the same was done by the closing of the Talmud. Therefore, an individual must be submissive to a beis din hagadol and cannot argue, so that every individual is bound to the mishna and talmud. It is true that the beis din hagadol who closed the talmud could have argued on the mishna, but they didn't.
The Rambam (end of hakdama to mishneh torah) seems to say it a little differently:
נמצא רבינא ורב אשי וחביריהם סוף גדולי חכמי ישראל המעתיקים תורה שבע"פ וכו' ואחר ב"ד של רב אשי שחיבר התלמוד בימי בנו וגמרו, נתפזרו ישראל בכל הארצות פיזור יתר וכו' אבל כל הדברים שבתלמוד הבבלי חייבן כל בית ישראל ללכת בהם וכופין כל עיר ועיר וכל מדינה ומדינה לנהוג בכל המנהגות שנהגו חכמים שבתלמוד ולגזור גזירותם וללכת בתקנותם, הואיל וכל אותן הדברים שבתלמוד הסכימו עליהם כל ישראל, ואותן החכמים שהתקינו או שגזרו וכו' הם כל חכמי ישראל או רובן והם ששמעו הקבלה בעיקרי התורה כולה איש מפי איש עד משה
The Rambam doesn't say that the closing of a talmud was a formal beis din hagadol decision that no later person can argue on. The rambam implies that since the chachmei hatalmud had a clear tradition, and after the sealing of the talmud the jews were dispersed in the galus so that the tradition was lost, one who would argue on the talmud would inevitably be wrong. It is assur to argue on the talmud because no one after the talmud had a tradition strong enough to measure up to the tradition of the chachmei hatalmud.