“So people just added certain statements and opinions of Amoraim throughout the generations?”
Added? Added to what?
Every Jewish boy studied the Mishna, by repeating it (with a tune) until he knew it well. There was no Sefer Mishnayos. (That’s right. If you want to argue, start a new thread.) Once they knew the Mishnah they would join a Yeshiva and discuss everything they learned. They would lesrn.from their rebbes what the previous generations had to say, and what the Halachah is. They would learn how to interpret and resolve certain hard Mishnayos and Braysos.
Then, they would argue and discuss new issues that came up, as well as weighing previously given interpretations. Those that rose to the top and became Roshei Yeshiva would teach all this old and new material to the next generation.
Obviously, students of different rebbes would land up with different interpretations and even, at times, different versions of earlier statements (or, more commonly, different attributions).
Eventually, Rav Ashi along with Ravina made one coherent Talmud, incorporating and ironing out everything that was taught until then. After him, the next generation committed it to writing.
The next few generations, the רבנן סבוראי, edited it to give it one language, to fill in steps to make it easier to follow, added some Sugyos (i.e. beginning of Kedushin) and stuck in a few explanations.
The generations following that are the Geonim, who were mostly occupied with Halachah. They also had many points that were passed down and weren’t written in.