Why was Louise the only one who advances her mind?

Louise has a unique understanding of the language.

Many of the scientists have worked out some of the vocabulary and seem to have made some progress in communication. But Louise has immersed herself more in the language than the others.

But the implications are not apparent until very late. In a final splurge of communication, done as the scientists start to panic about the reaction of the other nations to the partial communications they have interpreted, the aliens send a very rich message: not one word but a whole screenful of words. Nobody can make sense of it until Louise has a moment of insight and can read it all and get the point of it. Only at this moment does the full intent of the language have an effect on her: and when it does it rewrites her sense of time.

She then uses this knowledge of the future to avert a potential catastrophe. We only see the implications of this in the postscript (which may be actually her memory of the future) where many of the players meet for the first time at a celebration of her book on the language and one of them triggers her memory of what she did to talk down the hostility.

The key point is that she is the first person to completely grasp the alien language and therefore the first to experience its ability to undo her brain's sense of linear time.

As I understood it, Louise was the only one who actually understood the alien language, rather than merely translating. Different shades of meaning can be missed by simply translating word for word (example: tool vs weapon). Every language has idioms; phrases that don't make sense literally, but the meaning is understood to those who understand the language (example: "raining cats and dogs"). Louise's ability to understand time out of sequence grew stronger in conjunction with her understanding of the language.

While Ian worked closely with Louise, his approach to the language was mathematical, and he didn't share her way of understanding it. As for the other scientists around the world, their misunderstanding of the alien language led to their preparations for war, and to the rogue Americans setting off the bomb.

The way I see it, we don't know that everyone else wasn't getting visions of the future. In fact, it seems like everyone gets shaken up by the experience of meeting with the aliens, so it's very possible that everyone was like Louise and just didn't tell anyone about it. What they were missing, though, was the extra explanation that the language itself was linked to seeing the future. That's not something most people would be able to figure out on their own.

Remember, too, that Louise was able to read a book she would write about the language. From this, she was able to gain an even greater understanding and mastery of the alien language and the ability to see the future. It's possible everyone else was only days or even hours away from a similar breakthrough. However, since Louise solved the problem for them, they didn't need to keep searching, and could have waited to read the book normally.

So, in summary, it was the knowledge that her visions were visions of the future, and the breakthrough of using those visions to read a book that didn't exist yet, that allowed Louise to pull ahead of the competition and save the day. It's possible the other camps were days or even hours away from similar breakthroughs, but of course we never actually get to see that.

(Also, maybe no one else actually read Louise's book)

She went into the ship, a part that no one else went into. She understands that the alien wants to give a gift to help humanity so that in three thousand years humanity can help them. The gift was communication; for sight into the future.
This is something no one else did, this is why she was the only one that could see the future. The aliens left because of the way they were treated by humans, otherwise they may have given the gift to more people.

After the Aliens leave, we know that Louise dedicated herself to teaching the language, presumably so others could choose to study and fully learn the Aliens' language and receive their "gift".

From all the flashes that Louise has, they are all of her future, never of her past. This could mean that others eventually learn the language, and then from then on, perceive their present and future in non-linear fashion, enlightened in the same way as Louise. Otherwise, if no one else eventually received the "gift", mankind would have gone back to its old ways after the Alien threat left. No lasting unification of our planet would have occurred.