SPOILERS COMING....... WATCH OUT!!!!!
The information that we're given is that somehow, human beings (not necessarily on Earth) survive long enough to discover the non-linear nature of time (or 5 dimensional nature of reality, which is presented as kind of the same thing), and endeavor to affect past events so that humanity on Earth will spare itself of the worse suffering to come.
It doesn't need to be explained how, where or when this occurs, just that it occurred somehow. It's not unreasonable to think that someone on Earth survives underground, or one of the many people sent out in off-world exploration survives and makes this breakthrough. All that's required is that more than one person survive, and continue breeding long enough for this breakthrough in understanding to occur.
I think the reason people are considering the ending paradoxical is that the film, for reasons of wanting MM's character to seem protagonist-like, seems to imply that he's the one who brings this about himself, which doesn't make sense. We see him suffering on Earth, going to NASA, leaving Earth, going through the wormhold, visiting various planets, and learning stuff along the way, none of which involves discovering FOR HIMSELF the non-linear nature of time or the 5 dimensional nature of time. So yes, it would be paradoxical to believe that he created the wormhole and the time rift to send a message to his daughter, but I believe this is set up for him by the beings (future humans) who have discovered this almost godlike ability to create conduits through space and time, yet can't direct affect the past themselves (because "signals" or information can be sent through time, but human beings can't simply visit the past physically and communicate directly) and so rely on MM's character to send messages to his past self and to his own daughter which will be interpreted in ways that will be understood (because of this kind of inside joke language they share, more code and so on) and allow people who exist in the past to make changes that will flow forward into the future and enable the plan to succeed in making humanity's course on Earth somewhat less awful than it would have been otherwise, and to finally send a larger number of people off the planet, as we see happening at the very end.
Again, I think the confusion arises because this is effectively a deus ex machina ending -- godlike intervention from unspecified offscreen future humans actually saves the day, and MM's character merely acts out the role he needs to play, understanding hints he figures out along the way. He and his daughter make the change happen but only because they're acting through a mechanism set up for him by others we never see, who have somehow survived long enough to gain this higher understanding of time and space.
Now, if you think the film presents no possible way for any human beings, even a few of them, to survive long enough to gain this understanding and set up the means for this understanding to be passed along, then this explanation won't really work.
