As I understood, they "discovered" the time machine, that made David able to go back to his 7th birthday, after watching a 10 year old video of him doing so.
But how could that video exist, if it was the very trigger to conceive the time machine and thereby the event shown in the video?
If David had not time traveled, he would have never discoverd the machine which allows him to time travel.
A classic time travel paradox. Can be explained by viewing the direction of time to be relative. If effects can cause causes and causes can cause effects, if time future and time past are both present in time present etc. all moments happens simultaneously at the same time, we only experience them in an order. It is a mind blow but the mechanics on the quantum levels opens such radical thoughts. Or said differently; if time can flow in both directions, the future can effect the past.
"Reality is merely an illusion, albeit a very persistent one" - Albert Einstein
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** I am normally not a praying man, but if you are up there, please save me Superman **
Another explanation is to see each of their travels as new or parallel timelines. So they are not jumping in their own lines, they are only crossing to other timelines. This explains some of the more radical differences, like the plane crash.
Of course this would effectively mean that each timeline is predestined and therefore there could be no free will, but since there are infinitely many of them, it will seem as free will.
By this explanation there is one timeline that invents a timeline-jump device. He then jumps to his birthday in another line, and another timeline have him destroy the time machine drawings to maybe a time machine that would never have worked in that particular timeline. It worked in another timeline. Notice his father is saying, "you figure it out???". Just like the father never figure it out himself, and yet the movie we see does not show any actual invention. He even says, that he just followed the blueprint... so maybe he jumped to a timeline and destroyed a time device that would never have worked.
... hmmm, but then why did he disappear?
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** I am normally not a praying man, but if you are up there, please save me Superman **
Parallel timeline