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It was told earlier in naruto Shippuden that if Naruto uses rasenhurikan often it can lead to damaging chakra in his hand but towards the end he uses it often and his hands are not injured. Why?

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It's explained here:

Naruto is able to overcome these flaws by learning Senjutsu. By coating the Rasenshuriken in Senjutsu chakra, its shape is maintained after formation, meaning that he can use it as a projectile and, as a result, that he no longer needs to worry about damaging himself.

I'm not sure if this was explained in the anime since I only read the manga so if it's not, I suggest reading the manga starting at Chapter 400, which I believe is the start of Naruto's training for Senjutsu.

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