He was dying, but used his last strength to release her.

I often see people say that Eugene had to wait, what would Rapunzel cured him, and then he had to cut off her hair. But who knows what would have happened next. After all, gothel could push Rapunzel aside, in order that Eugene did nothing.
But instead, he made a selfless act, at the first opportunity cut her hair. But that is not the point in this scene, and the fact that Eugene was selfish, because he grew up in poverty and in the shelter, so he had to be selfish to survive, so he turned to theft. But this time it was different, at this point he puts the interests of others above their own. Love Rapunzel and Eugene are so strongly reminiscent of the story of Beauty and the Beast, which is what makes this pair one of the most amazing in the entire history of Disney.
Secondly, when Eugene cuts hair it is said that there is someone who loves Rapunzel, and not her hair. Eugene fell in love with Rapunzel, he fell in love with her heart. That's why he cut hair, they don't matter to him.