My players in general are not much into RP, they don't pay much attention to the story, they don't take notes. They like to skip stuff, like "we're there yet?", "I tell him everything I know, I'm there yet?". Like that

And, one of them got the Keen mind feat just to f*ck me up. It's basically another mental load he will put on me because now he have an excuse to not take notes and not pay attention to anything, and more, he can get to the NPC and play the card "I tell him everything I know".

Anyway, how to handle that?

Better yet, how to motivate the players to get more involved with the story I'm put so much work on?

PS: The idea of including their backstory into the plot will not work since they backstory is like "my parents are dead and I don't remember anything"

Your problem is less Keen Mind, and more a mismatch between what you want out of the game and what the players want. They seem to want less detail, and presumably more action, than you do. This is not a problem that can be solved without talking to your players about it. You may be able to reach a compromise that everyone is happy with. Or you may not.

As for the feat itself, this is what it's for: to let a player who can't or doesn't want to track a lot of info not have to. You can't, or shouldn't, stop it from doing what it's supposed to. In the particular example, I don't see what the problem is: infodunpy conversations like that are really hard to make interesting, especially when there's no plan to deceive. There are conversations like that ("I tell the authority figure everything I know about the evil cult." "OK. They look thoughtful, and thank you.") all the time in my game, and my players are engaged in what's going on.

In my current game information is very important, specifically what someone chooses to reveal opposed to what they keep secret. The party have gotten themselves into several dubious situations where some of them have acted immorally. Whenever a player comes to the fore with "I tell them everything I know" I have the response of "When you get to the part about killing the prince their eyes widen with shock." Even something like revealing that they discovered magical artifacts becomes unwanted because the government in the game believes that such tools must belong to the people and aid in cities' defense.

If your party is very virtuous, perhaps you can instead have the NPC's that deal with the party be more suspicious and twist or share information in ways that harms the party.

If you really want to weaponize information it needs to be of the nature that it holds sway over someone's fate. E.G. Evidence in a court case where the death penalty is a possibility. Troop movements of armies that can easily wipe out a city.

However if you're running the average treasure hunter game it will require a lot of changes on your side. Perhaps there is a competing group of treasure hunters and the location of the next dungeon is information worth keeping secret.

I simply banned this feat for exactly this reason. If they don't want to write something down, that on them. It's actually the only ability from any book I have banned.

Keen Mind is the Sharpshooter/GWM equivalent feat for Wizards. It’s supposed to be strong.

It sounds like your players aren't into exploration, and Keen Mind is largely an exploration feat. I would suggest talking to the player who took the feat about taking a different one. The Keen Mind feat apparently doesn't fit well with your game's hack-and-slash style, and you should bring that up. 

Another note, maybe you should put less emphasis on the plot of your story. This player clearly took the Keen Mind feat to avoid getting involved in exploration and roleplay they aren't interested in. Just let your players hack happily through monsters instead of feeling like you need a plotline to impress them.