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", Donnie is like Groundhog Day's character Phil minus self-aware, plus MD-aware).
When the engine from the Oct. 30 1988 PU drops (into the past), the PU is gone, the TU is created. Similarly, when the engine in the Oct. 30 1988 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 and specifically how he saves the universe.
At the introduction, 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 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. He is finally successful in ending the looping time paradox by preventing the Jet-engine from landing where it already exists.
There are many instances that show Donnie Darko is caught within a repeating time-loop and has done it all before. This is explained in the PoTT Dreams chapter, where it states the manipulated living will have some recollection of past events.
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. This explains why the PoTT states the universe will be "destroyed".
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 or remembers something about Frank's injured eye (that Donnie shot in the prior 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 of Donnie's recognition and awareness of past events.
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) Donnie and Mrs. Farmer's class are watching the Jim Cunningham video, and Frank says to Donnie, "Pay close attention, you might miss something."
7) 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-Bunny 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. 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 one explanation 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).
8) 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.
9) After Donnie's first encounter with Frank on the golf course, Frank does not reveal his name (not until they are in the theater), yet, when he is with his therapist he tells her he made a new friend, Frank.
Stopping this time-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 prevents the time-loop from continuing and therefore saves the future.