WHAT HAPPENS, AND WHEN?
October 1, 1988. Donnie is on the hill, looking at the sky, laughing. He rides his bike home and goes to sleep.

October 2, After midnight, Donnie is woken up by Frank-Bunny, saving him from a jet-engine crashing into his room. The jet-engine is actually from October 30, disappearing through a time corruption 28 days into the past. This catastrophic event distorts spacetime and creates a Tangent universe (TU). In the Tangent Universe, Frank-Bunny guides Donnie Darko on a 28 day journey through this alternate tangent dimension to save the universe:

October 30, Tangent universe: After his journey through the TU, which we witness in the movie, the jet-engine breaks off the plane (that his mother and sister are coincidentally on), and from the Tangent universe, Donnie using a wormhole sends it back to the Primary universe (PU), giving it a reason for being. The only object that went back in time from the tangent universe to the primary universe was the jet-engine. In doing so the tangent universe and everything in it is replaced by the primary universe, and time is reset to:

October 2, Primary universe: Jet-engine appears in the Primary Universe (without explanation to everyone but the viewer), Donnie and everyone else are reset and repositioned. Primary-Donnie, back in bed, is killed by the Jet Engine.

HOW DOES IT HAPPEN?
The theatrical release of "Donnie Darko" offers very few answers. However, the directors cut DVD is much clearer, with specific explanations about events taken from the book "The Philosophy of Time Travel", written by the character in the movie Roberta Sparrow aka Grandma Death.

The Jet-Engine from the future is the Artifact: "If an Artifact occurs, the Living will retrieve it with great interest and curiosity. Artifacts are formed from metal"

Donnie Darko becomes the Living Receiver: "The Living Receiver is chosen to guide the Artifact into position for its journey back to the Primary Universe." Donnie is chosen because he is the person most affected by the Artifact, hence he is the center of the tangent universe and the only one who can close it. Donnie's job is to return the artifact to the primary universe. "If the Ensurance Trap is succesful, 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. The paradox that must be created is Donnie sending the engine back in time through a wormhole, to prevent the other paradox, a second jet-engine in the tangent universe.

Frank-Bunny is the Manipulated Dead: "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."

Returning the Artifact: At the focal point in the paradox as the tangent universe winds down, a wormhole (a tear or hole in spacetime) appears. It is at this moment that Donnie must guide the jet engine out of the tangent universe, as we see succesfully preventing the paradox for the first time. Otherwise this tangent is destroyed, reset, and the cycle repeats.

When the Tangent universe is reset: "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".

WHY DOES IT HAPPEN?
The movie "Donnie Darko" is about Donnie's first successful quest in ending the temporal paradox cycle or "time-loop" which he is caught in (not unlike Bill Murray in the movie "Groundhog Day".)

When the engine from the Oct. 30 PU drops (into the past), the PU is gone, the TU is created. Similarly, when the engine in the Oct. 30 TU drops (when the LR/Donnie fails), the TU collapses, and the PU is reset, at the point we see in the beginning of the movie. This is the loop, and it continues until the engine does not drop in the TU because it is sent back to the PU by Donnie. Ending this repeating time-loop is directly how he saves the universe.

At the intro, Donnie is laughing as if he remembers something, specifically his own survival in the previous tangent loop, when looking at the sky. This is where the movie begins, and it is also where the tangent is reset and the loop begins when Donnie is NOT successful. Then the loop continues as we see when the engine drops into the past via a time corruption to create the tangent universe. On that previous previous quest , Donnie did not send the artifact back. When he does and the time loop finally ends, he dies alone as he always feared, but with a sense of completion and relief hence the laughter and the accomplishment of saving his family and friends and the entire universe because he is finally successful in ending the looping time paradox by preventing the jet-engine from landing where it already exists.

when the time loop finally ends, he dies alone -as he always feared to- but with a sense of completion and relief (hence the laughter) and the accomplishment of saving the world, along with his family and friends.

More evidence that Donnie Darko is caught within a repeating time-loop.

1. In the classroom, Donnie states: "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." When the world is torn apart in 28 days, when Donnie fails, it is destroyed but then re-created, reset. This is the time-loop, this form of "destruction" when Donnie fails is not unlike the "destruction" of the jet-engine in the PU, re-appearing in the TU (or vice-versa). On a larger scale, the destruction of all current existence, by being repositioned and reset in a different timeline, specifically the Primary Universe when Donnie is laughing on the hill looking at the sky.

2. When the engine crashed through the house, an alternate universe had been created by the falling jet-engine; however Frank-Bunny appeared BEFORE the engine crashed, before the Tangent Universe was created; therefore Frank-bunny must have traveled from another point in time ie a previous tangent universe in the repeating cycle.

3.In his bathroom mirror, Donnie sees Frank and starts stabbing Frank's eye through liquid. Evidence that Donnie knows / remembers something about Frank's injured eye from a previous TU.

4. In the theater when Frank-bunny has taken of his mask, Donnie asks what has happened to his eye. Frank turns his head and stares, and Donnie looks at the screen and starts to smile. Another clue that Donnie has done all of this before in the previous timeloops.

5. Also in the theater, Donnie asks Frank about his name, he responds with "Its the name of my father, and his father before me." another implication that they are separate entities but connected/related.

6. The only way Frank could have been killed by Donnie at the end of the movie was if Frank-Bunny had already saved Donnie at the beginning of the movie. The Manipulated Dead Frank-rabbit appears before he himself is assumed to be created when Frank is shot by Donnie, it follows that this Manipulated Dead Frank-rabbit could not have come from the Frank seen in the movie, but the Frank, shot by Donnie, in an unseen previous Tangent Universe. Clearly, this was not the same entity, as an existing thing cannot create itself for the first time. - see http://www.imdb.com/title/tt0246578/board/inline/28323929 - Each different Manipulated Dead entity has expired in the FAILED quest to guide Donnie to "use his Fourth Dimensional Power to send the Artifact back in time into the Primary Universe before the Black Hole collapses upon itself." Frank had to die in a previous tangent before coming back to rescue (and guide) Donnie. Since the tangent universe is eventually destroyed by being reset, this is how Frank initially died and then returned in the next tangent universe as the Manipulated Dead and guided Donnie to eventually shoot him (part of the Ensurance Trap set for Donnie to send the engine back).

7. The repeated time loop theory also explains why many of the central charaters in the film (Mrs Pomeroy, Gretchen, Grandma Death, Dr. Thurman), all seem to have some kind of inside knowledge and a sense of what's going on as though they are all merely acting their lives as in a well rehearsed play, especially Grandma Death and her endless trips back and forth to the mailbox looking for Donnie's letter, and Mrs Pomeroy bringing Gretchen and Donnie together and writing "cellardoor" on the blackboard.

Stopping this looping cycle revolves around the free will of Donnie Darko. As the Living Receiver in the Tangent Universe, he will either be motivated and learn enough to be able to choose to send the engine back to the past, which closes the loop and prevents the engine from falling on October 31, or he will fail and the time loop will restart itself, the jet-engine will fall. With the evidence that this has happened many times previous, the Manipulated Dead has failed in all previous attempts to prevent the paradox, failed to get Donnie to send the engine back, failed because Donnie would either not survive long enough to send the engine back, or he would choose to not send the engine back knowing it would mean his own death. The Manipulated Dead and the Manipulated Living persuaded Donnie in many ways: Grandma Death (Sparrow), Dr. Monnitoff who gives him the Philosophy of Time Travel and explains his paradox theory with "chain reaction that could destroy the universe", Frank-Bunny, meeting and becoming close with Gretchen, Frank killing Gretchen, everything was part of the ensurance trap, so Donnie would a) know how, know why, and want to change things so that Gretchen will live again. Donnie finally realizes enough to prevent the paradox from taking place by sending the engine back. This time the universe resets to the PU with Donnie in his bedroom.

Donnie laughs for the same reason the other characters were reflective at the end. As explained in the POTT, they have some memory, if not of details than of a feeling. And since Donnie saved the world by finally preventing the recurring timeloop, it makes sense he feels great (even if he doesn't understand why, and especially since he doesn't realize he is about to die).

After the engine crashes and kills Donnie, the OTHER (identical) engine that already exists in that timeline will be investigated thoroughly, and will not fly, or crash, on this October 30th timeline, hence no tangent will be created, the cycle is over, that is how Donnies action saved the future.