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I remember this story being told to me about this alien race that observed this planet and its intelligent life. They find out that the planet will soon die due to the sun or something and want to send a rescue mission to save the people.

Turns out the people have already figured this out and saved themselves. The plot twist is that this planet is actually earth and the civilisation is the human race.

I’ve been trying to find what this story is but I haven’t found anything. Does anyone have an idea? I believe it isn’t an incredibly long story. Nor is it pessimistic — it’s actually a very optimistic story of the human race.

"Rescue Party", a novelette by Arthur C. Clarke, also the (unaccepted) answer to this old question. First published in Astounding Science Fiction, May 1946, available at the Internet Archive. The full text and a reading are available at Escape Pod.

Wikipedia plot summary:

The story begins with a ship full of aliens visiting Earth only hours before the sun will explode, destroying the planet. The mission of the aliens is to try to save as many people and as much of the culture as possible. Normally the galactic civilization does surveys of planets every one million years for new species, but the human race did not exist the last time the survey was done – four hundred thousand years before. However, radio signals had been detected on a planet 200 light years away, indicating intelligent life had arisen.

To the aliens' surprise, the planet seems to be empty of intelligent life, except for the remnants of a civilization. While the aliens explore the old cities, we find out that it is typical for species to take thousands of years to develop from radio to space travel.

During their exploration of Earth, the aliens find a communication tower beaming into space. At the end of the story, they follow the beam and find an enormous fleet of human ships, powered by rockets. The aliens, equipped with faster-than-light ships, are amazed that humans dared to cross interstellar space with rockets, having done so only 200 years after inventing radio. Humans do not possess faster-than-light ships, but in order to survive as a species have taken the audacious tactic of using generational starships, in the hopes that their descendants will one day arrive at a new planet.

The final lines of the story have the aliens musing about the nature of human civilization and the future of the humans when they learn about the existence of other intelligent life, given their rapid advancement and apparent determination. The last sentence hints the outcome was not favorable to the aliens.