So, I beat the game a few days ago after having bought it and getting to the last save point back when the game came out. I didn't know I was at the very end of the game until I pulled up a guide (I was determined to finish the game after having it sit there for so many years) and I was glad to find I didn't need the guide at all.
 
Anyway, after the eight or so year hiatus I was obviously fuzzy on some of the plot. The ending didn't make nearly as much sense as it probably would have if I beat it back when I had played through the rest. So, I have about three questions for anyone who knows.
 

1. Why do you have to kill the Aeons? I can't seem to remember at all anything about the story talking about this.
2. Why does Tidus start to disappear? Is he dead like Auron (I assume Auron was dead the whole time.)?
3. What was Yu Yevon, and why was Jecht Sin?

I'm sure my answer isn't perfectly accurate, however it's all connected.

 
You need to kill all the aeon to get the final summoning. When that happens, you can defeat sin, however the best way they found to do that is to sacrifice a life of a summoner which replaces the soul of sin or that's what the summoners were told which isn't completely true. It's just a ruse for yevon ppl to stay in power and to promote their religion as a controlling interest. Ject was the one who decided to sacrifice himself and they always done it that way because it was really the only way to save spira as they were told. From wiki, sin was created by Bevelle’s enemy, Yu Yevon, a Zanarkand summoner, as a means of destroying Bevelle.Yuna changed all that when she didn't want to sacrifice anyone. Yu yevon is someone who summons, I can't remember if he was part of sin, or the other part of it. The faith and those spirts were dreaming, tidus and auron dissapears because they were only part of a dream. The real tidus in FFX-2 was called shuyin and he's not nice at all.  
 
I can't say it's entirely accurate, however anyone else can add or elaborate on it.

I'll try to put this the best I can. We'll start with Yu Yevon, who is the easiest character to start with for your answer. Yu Yevon was the leader of Zanarkand before it was destroyed 1000 years prior to the game. When he realized how boned his beloved city was due to Bevelle's machina, he took his summoners and they created the Fayth. There are the same Fayth you see in the temples and the final area of Mt. Gagazet before the Zanarkand ruins. When the game comes around, you can essentially consider Yu Yevon as the ultimate Fayth - his aeon being the abomination Sin.
 
As Yu Yevon was controlling Sin, the remaining Fayth either became the Aeons that you see in the game, or in the case of those in Mt Gagazet's wall - they were dreaming up Tidus' Zanarkand. It's assumed that "Dream Zanarkand" is somewhere far away from the normal world of Spira - at least in a place where it can be accessed by Sin. We never truly know why Jecht gets transported to Spira, but he does. To answer your second question, Jecht became Sin when Braska used him to create the Final Aeon - his form you fought as the "final" boss in the game. The creation of an Aeon so powerful required the strong bond between a summoner and a guardian - in this case Braska and Jecht.
 
The main reason you kill the Aeons is due to the fact that one was required for Yu Yevon to create a new Sin from it. With the Final Aeon dead, his next best option are your standard Aeons, so they need to die in order to prevent Sin from being recreated. When the Aeons and Yu Yevon are both defeated and sent, there's no reason for the Fayth to stay alive and keep their Dream Zanarkand around. When they stop dreaming, everything related to it disappears - including Tidus.

1. Why do you have to kill the Aeons? I can't seem to remember at all anything about the story talking about this. 
 
Yu Yevon is the being that takes over the final summoning after sin is defeated every time to create the new sin. They work out that every time one of the summons is being controlled by Yu Yevon and is defeated by another at your disposal, it jumps bodies to the one that just killed the aeon it was possessing. Basically you are attempting to give Yu Yevon no where to run to.   
 
2. Why does Tidus start to disappear? Is he dead like Auron (I assume Auron was dead the whole time.)? 
 
The fayth are some of the last summoners of Zanerkand after the 'summoner war' 1000 years ago that resulted in Zanerkand's demise. Whilst the 'dreaming' fayth exist (which are just the fayth), the memories of people from the 1000 year old Zanerkand exist, but by the end of the game when the summons are destroyed and the fayths are no longer needed they stop dreaming, and tidus is part of the dream.
 
3. What was Yu Yevon, and why was Jecht Sin? 
 
Yu Yevon as already mentioned is the being that jumps from Sin to the 'final aeon' each time Sin is defeated, in order to create the new Sin, thus why Sin can never actually be defeated and there are only calms. Lord Braska was the last to beat Sin before Yuna and his final aeon was created from Jecht who gave his life (as anyone who becomes a final summoning must do, and is usually a member of the traveling party with the summoner with which the summoner has a close bond, so it could have been auron instead even) and so when the final aeon for braska (jecht) beat sin, he became the new sin.