Story

Five Nights at Rusty's 1
The story of Five Nights at Rusty's was revealed in a news article on it's official gamejolt page; here

'As I mentioned, one thing that incredibly discomforted me is how closely the game paralleled Five Nights at Freddy's. I had a story and gameplay planned out for possible sequels, if the game was successful (The FNaR 2 from April 1st was a prank). However, none of them felt original, or like my own.

The most original feeling one was 4, because it didn’t follow some sort of ‘kid is hurt by animatronics gets nightmare hallucinations of them’. But even with it being a bit more original, was it even needed?

The entire story of FNaR can be told in one game. The Toy Designs (Toy...Toy Designs?), ghosts and nightmares, were just added to parallel FNaF.

The story of FNaR was going to be that you play as Lyric Musille (a character I will be taking out of FNaR and using in my own original works), a young teen working at Polaris' Pizza Place, a reboot of Freddy Fazbear's Pizza; with the animatronics being repainted into the new ones.

Polaris was a repaint of Toy Freddy, Pina a repaint of Toy Chica, and Coriander a repaint of Toy Bonnie. (This is one of the reasons I feel that the trio weren’t meant for a fangame. PART of their backstory is they’re literally just repaints of the canonical characters) In addition, Rusty was a redesign of the FNaF 1 Freddy Fazbear.Even the pizzeria itself was a slight extension of the FNaF 1 pizzeria. The similarities were all part of the plot but.. It still made me unhappy and I didn’t want to rewrite the plot.

Throughout the game Phone Guy would tell you about the animatronics freezing up and breaking down during the daytime. Eventually, Lyric gets their pay check on Night Five.'

Five Nights at Rusty's 2
'In FNaR 2, Lyric would have come back for another week in pursuit of info about the missing guards. The old animatronics have been replaced by the…Toy...Toy animatronics. Stuff would have probably happened.'

Five Nights at Rusty's 3
'In FNaR 3, Lyric comes back one final time as they try to piece the pieces together. Rusty is the only animatronic left however he would rebuild the broken pieces of the animatronics, which would awaken the ghosts of them.

I don’t have the concept art for it because I lost the notebook I had it in but the game would have had 360 degree rotation, and it would have been possible for Rusty to attack you from behind.'

Five Nights at Rusty's 4
'And in 4, the building will have been upgraded into a more scary horror attraction, complete with new, nightmarish animatronics. A new animatronic joins the party, a Bonnie shaped mess named Hybrid.'

Backstory
The backstory would've been that the four security guards, Jeremy, Mike, Fritz and the original Phone guy, were murdered during their shifts and stuffed into the four suits. Jeremy went into Polaris, Mike into Coriander, Phone Guy into Pina, and Fritz into Rusty. They would've gone after you because they falsely believe that you are Purple Guy, and want you to suffer with them.

There are some things that the creator says that he 'forgot', but we'll see.

Trivia

 * On Night Six, they leave with some overtime and information on four missing employees.
 * Completing Night Seven would have rewarded you with an unknown prize.
 * And completing Night Seven, with all the animatronics set to 20, would have rewarded you with another unknown prize.