I don't exactly understand what happened with Fry going back in time and staying there, when he became Lars. We also see that he stayed in the future. Could someone please explain how this is possible?
Fry travels back in time to January 1st 2000 and finds I.C Weiner's pizza. The pizza is cold so he creates a duplicate of himself (Lars) to get it when it was warm.
The duplicate goes back to working at Panucci's Pizza and then decides to rent the apartment above the shop. He work there for 3 years when he decides to get a job helping the sad narwhal Leelu who wasn't eating.
Fry was the only person who could make the narwhal eat and they become very close.
In 2010 he has to set Leelu free into the sea. Fry was missing Leelu so much he hires a ship to set out and find her. After 2 years of searching he finds her but she wants to be free with the males narwhal.
Fry returns to he apartment where bender tries to kill him. Fry's hair caught on fire and he lungs were poisoned by the smoke. This is when he discovered that he was Lars Fillmore.
He quickly goes back to the Cryogenics Lab were he freezes himself in Michelle's tube for 990 years. He wakes up in the year 3002 with his new name and gets a job at the Head Museum where he meets Leela when Hermes is put into a head jar.
Hermes gets his body back with a time paradox duplicate. The professor explains that these duplicate are doomed to die.
Lars and Leela are about to get married when Hermes's body was crushed. Lars realises that he too is a time paradox duplicate so he calls off the wedding.
A duplicate bender self destructs killing the Nudars and Lars.
The timeline is a bit messy, but in essence, there are two Frys, whom we shall call "Fry" and "Lars".
Lars split from Fry, becoming a duplicate, when he chose not to go back in time for pizza after defeating Bender, and ended up staying in the past.
Fry, on the other hand, ate the pizza, and then hid in his own tube, returning to the future.
There are more Frys on the board at certain points, but they are all pre-split Frys, and are all on the path to become the real (non-duplicated) Fry.
Lars' refusal to toy with time is, hilariously, what causes him to become a paradox time-duplicate in the first place.
Fast-forward 12 years, and Bender burninates Lars' apartment, making Fry realize that he was Lars all along, and causing him to rush off to travel forward in time to be reunited with Leela.
Edit: As a clarification: No version of Fry stayed in the future; but the main timeline Fry traveled to the past and came right back.