Firstly this explanation is written assuming that you've seen the film. There are spoilers throughout so it could ruin the enjoyable experience of watching it for the first time.
The film has been carefully made so that there are multiple explanations to what happens. With the release of the director's cut some of the ambiguity has been taken away and the director's true version of the story becomes clearer. The explanation heavily revolves around the book Donnie gets from his teacher, "The Philosophy of Time Travel" written by Roberta Sparrow (Grandma Death). The book is a fictional one but some of the pages were in the film's web site and included into the director's cut version of the movie. You can read those pages here.
PU = Primary Universe (The Universe we exist in now)
TU = Tangent Universe (The parallel Universe that most of the film is set in)
PoTT = Philosophy of Time Travel
The PoTT states that time is usually a stable construct but every now and then the fourth dimension gets corrupted. When this happens it creates a TU which is highly unstable and will only last a few weeks before it collapses in upon itself. There is a danger that when the TU collapses it could cause a black hole capable of destroying the PU as well.
This is what happens in the movie, there is a corruption in time and at midnight on October 2nd a TU is created. The next 28 days are now set in this alternate reality. Shortly after the TU starts Frank wakes Donnie up and lures him out of the house. A few minutes later a huge jet engine falls though the rip in time and lands in Donnie's bedroom.
It is important to remember that the TU is NOT created by the jet engine or Frank waking up Donnie. We are already within the TU when both those events occur. Unfortunately we never find out what caused the TU to begin, it's just an unexplained phenomenon.
According to the PoTT when a TU is created an Artifact will also spontaneously appear. The Artifact is the first sign of evidence that a TU has been created and are always made from metal. In this case the Artifact is a giant jet engine.
Where the Artifact originally comes from is never disclosed. It was most likely created as the TU was being formed. The TU is an exact copy of the PU with one of everything copied into it. During the copying process there was a glitch and two jet engines were formed. The second engine has nowhere to go as it already exists so it's just dumped where the TU began, over Donnie's house.
The Artifact makes the TU unstable; it can't handle having the anomaly of a duplicate object within it. In order for the TU to unravel without forming a black hole it must once again be an exact copy of the PU. By removing one of the duplicate jet engines it will balance out the Universe allowing the TU to collapse safely. The only way to remove the Artifact is to send it through a portal into the PU.
When a TU occurs with an Artifact then a Living Receiver is also chosen. This person is chosen seemingly at random and in this case Donnie is the chosen one. As Donnie was at the epicentre of the TU this is most likely why he was selected. The Living Receiver's mission is to guide the Artifact out of the TU.
This person is usually blessed with some supernatural powers during their time in the TU to help them with their quest. Some of these include increased strength, mind control, the ability to conjure fire and water and telekinesis.
Donnie is not aware of his responsibilities at first though and he doesn't even know he is within a TU. As his journey continues he slowly starts to realise what is going on and what he must do.
Anyone connected to the Living Receiver who dies in the TU becomes Manipulated Dead. These people are very powerful and have the ability to move through time and talk to the Living Receiver through a Fourth Dimensional Construct. They also have some knowledge of the impending disaster and help guide the chosen one.
Frank is killed in the TU and therefore becomes one of the Manipulated Dead. He travels back in time and helps Donnie with his quest to return the Artifact. The first thing he must do is save Donnie by waking him up before he gets squashed by the jet engine. He is like a reverse ghost who appears before he was killed. Gretchen is also Manipulated Dead but she doesn't appear to Donnie in the same way as it might be to confusing for him.
All the people connected to the Living Receiver are the Manipulated Living and these people will subconsciously help guide the Living receiver. They will behave in the exact perfect way necessary to push Donnie toward his eventual destiny. Nearly every event in the film has a specific purpose designed to aid Donnie to save the Universe.
There is a driving force behind the Manipulated's behaviour... although we never find out what it is for sure. God is one of the most popular ideas mooted, the film does use religion at many points. The story uses lots of religious symbols and often points to the idea that you follow God's channel to decide your own fate. This guiding force is moving all the Manipulated characters around like pieces on a chessboard, making a sequence of events that will lead to Donnie's success. The Universe is in grave danger and this guiding force has stepped in to help save it.
The PoTT states that all the Manipulated are trying to lure the Living Receiver into a trap so has no choice but to send the Artifact out of the TU into the PU. The PoTT calls this the Ensurance Trap (old English) . This is effectively the role all the Manipulated play, they are trying to save themselves by guiding Donnie to complete his mission. If Donnie fails they die as well.
Frank tells Donnie to flood the school which in turn leads to Donnie walking Gretchen home and him then asking her to be his girlfriend. This event is not just coincidence; it's specifically put in place as part of the trap.
Gretchen plays a crucial role in the trap, remember Donnie only meets and falls in love with her within the TU. He is given someone to love then she dies. He's so upset at Frank for killing Gretchen he shoots him dead. This has now created Manipulated Dead Frank, the person guiding him to do these things. He must create Manipulated Dead Frank in order to be saved himself in the beginning.
The manipulated have now successfully set the trap. Donnie was saved from death and is now in a desperate situation. His girlfriend is dead and he's a killer on the run. He knows what his purpose is now and only has one way out even though it will ultimately end his own life. Sending the engine through a portal will erase the last 28 days from history and time will return to October 2nd... back to when Gretchen was still alive.
At the end of the film we see a vortex appear over Donnie's house. It's the beginning of the TU collapse and is centred over where it began. Donnie is aware of what he must do now so he drives up the mountain for a better vantage point. The plane with the same jet engine has been manipulated to fly overhead at this exact time. Donnie rips the engine off using telekinesis, constructs a time portal from water and guides the engine through it.
We now see parts of the last 28 days rewinding as the TU starts to unravel. The TU has collapsed safely and the PU that was on pause starts back up from when it stopped. We are now back in reality on October 2nd, the last 28 days never happened. Donnie wakes up in his bed laughing after dreaming some of the events within the TU. He then goes back to sleep seemingly content with life now. The experience has seemingly brought him closer to God and he is no longer afraid to die. The jet engine Donnie sent through the time portal now falls into his bedroom killing him.
Even though the last 28 days never actually happened some of the Manipulated will be haunted in their dreams by their experiences within the TU.
There are bits of evidence to prove this. Frank touching his eye in the final montage and Gretchen waving to Donnie's mum proves that some of the characters have some knowledge of what happened within the lost world.
We see Jim Cunningham crying in the montage as he wakes up... clearly disgusted with himself. Ten days later he clears out his child pornography dungeon and shoots himself on the 14th hole of the golf course... his dirty secret is never revealed.
Many people assume Donnie has to die otherwise the TU would start over again. From a scientific point of view Donnie's death is not relevant at all , the corruption is already fixed before he dies and if he got up and avoided death then nothing would change. The PoTT tells us of previous Receiver's who died after the Artifact was returned but we only hear about them because their death defies logic. Any Receiver's who escaped death would go completely unnoticed.
It's almost certain Roberta Sparrow was a Living Receiver in a previous TU and that is how she came to write the PoTT. She survived her experience proving that the Receiver doesn't always have to die.
This film does blend science and religion together so his death may emphasise him as a saviour and his belief in God. In death Donnie's role as the Living Receiver seems even greater than before.
Other common questions concern the Artifact Donnie sends through the portal at the end of the movie. Many people ask why the PU is not in danger now it has a duplicate object with it. It seems to be the case that the issue of duplicate objects is only a concern once the Universe they're within starts going through the collapsing process. The PU is a stable linear time-line that never goes through a collapsing process so is therefore never in any danger from duplicate objects.
As stated above the Arrtifact is not the cause of the TU, it's just a by-product of it. This is why there is no danger of another TU starting up once Donnie sends his Artifact through the portal at the end of the movie.