I was confused at first too. When I finally got it, I wrote everything down. I think it's better if you read all of them :)
According to the book "The Philosophy of Time Travel" written by Roberta Sparrow (Grandma Death who keeps checking the mail):
1. "If an Artifact occurs, the Living will retrieve it with great interest and curiosity. Artifacts are formed from metal"
Jet Engine is the “Artifact”. It crashed into Donnie’s room on 2nd, October. Jet-Engine is actually from October 30 (the one his mother and sister are on). Therefore creating a Tangent Universe (TU) or alternate universe. The movie is about Donnie’s 28 days journey in TU. TU will destroy itself in 28 days. That’s how the world is going to end.
2. "The Living Receiver is chosen to guide the Artifact into position for its journey back to the Primary Universe."
Donnie is the “The Living Receiver”. He has to return the artifact to the primary universe.
3. "If the Ensurance Trap is successful, the Living Receiver is left with no choice but to use his Fourth Dimensional Power to send the Artifact back in time into the Primary Universe before the Black Hole collapses upon itself.”
Donnie has to create a wormhole and send the Jet Engine back to PU. If he succeeds, TU will be closed, putting PU back to its original state.
4. "The Manipulated Dead. If a person dies within the Tangent Dimension, they are able to contact the Living Receiver through the Fourth Dimensional Construct...The Manipulated Dead will often set an Ensurance Trap for the Living Receiver to ensure that the Artifact is returned safely to the Primary Universe."
Near the end Frank/Rabbit is shot by Donnie and this makes Frank “The Manipulated Dead”. In the beginning Frank/Rabbit saves Donnie so he won’t be crashed by the Jet-Engine because Donnie is the The Living Receiver and he alone can save PU.
When TU is created by the artifact (Jet Engine), the PU doesn’t exist anymore. If Donnie fails to create a wormhole to put Jet Engine back in PU, TU will be destroyed on 30th, October. Then PU will be reset back to October 2, creating a loop, Frank will warn Donnie again, Donnie will try to return the artifact. Donnie’s been stuck in this loop for a while. Remember he was lying on the road in the first scene? He failed to return the artifact and PU was reset. Donnie said in the class “"Well, earlier in the book, they say destruction is a form of creation. So the fact that they burn the money is ironic; they just wanna see what happens when they tear the world apart." It means destructing TU only resets/creates PU.
5. "When the Manipulated awaken from their Journey into the Tangent Universe, they are often haunted by the experience in their dreams. Many of them will not remember. Those who do remember the Journey are often overcome with profound remorse for the regretful actions buried within their Dreams".
Donnie was laughing and looking at the sky when he woke up on the road. Meaning he remembers a little. Now he has to try and return the artifact again. In these loops Donnie always send a letter to Grandma Death. That’s why she’s always checking her mail. She also remembers something. Frank’s ghost is not from the future. He was created in the previous loop where Donnie killed him and failed to save PU. Frank said in the theater when Donnie asked about his name: “It’s the name of my father, and his father before me". It means there were other Franks before him. Donnie also remembers doing something to Frank’s eye. He keeps stabbing Frank’s eye in the mirror.
6. “Ensurance Trap”
It is set for Donnie to complete the mission. Throughout the movie Donnie is guided by Frank/Rabbit. Other people are also helping Donnie but they have no idea they are helping him (or) they do remember, better yet, having a feeling/déjà vu to do these little things. Dr. Thurman gave Donnie “The Philosophy of Time Travel". Teacher Karen Pomeroy makes Gretchen sit near Donnie, (Donnie mentioned he doesn’t want to die alone) making Donnie fall in love with Gretchen so when she dies, he wants to save her by sacrificing himself). “Ensurance Trap” is like a god’s plan for Donnie to save the universe.
In the end Donnie stops the loop and successfully sends the Jet Engine back to PU. Everything is reset. Donnie wakes up on 2nd October in PU and crashed by the Jet Engine he sent. Gretchen doesn’t know Donnie in PU but she has a feeling like she knew him when she sees his dead body.
This is just my interpretation... I'm sure other people interpret this movie differently :)
The movie didn't mention anything about why sending the artifact back to PU will set everything straight... It just says it will... So I can't really explain it more...
When Donnie saved the PU by sending the Jet-Engine back, that Jet-Engine is from the TU. His mother and sister's plane won't crash cuz Donnie is dead in PU (he is the one who crashed it in TU). It's a loop lol So yes, Jet-Engine has no existence in PU.