When Naruto first used Rasenshuriken against Kakuzu the effect was massive. He destroyed two of Kakuzu's hearts, but there were a huge after effect to him. If I'm not mistaken, Tsunade advised Kakashi to ask Naruto never to use the technique again due to the huge risk. So why is Naruto still able to use the technique and have no side effects after that battle? I may have missed out on something.
Naruto only used this jutsu one time in normal form (against Kakuzu, and got many side effects, like you said), the other times he was at Sage mode, at this state he didn't damage himself.
According to Naruto Wiki:
Naruto later improved the Rasenshuriken with Sage Mode. This allows him to throw it at his opponents, and removes the threat of damaging himself with the technique.
To expand on Rikkin's answer, the damage is only done on impact.
When Naruto first tried to use it on Kakuzu, his arm was at the very center of the damage radius when it impacted, which caused the damage.
While training for Sage Mode, Naruto learnt to throw the Rasenshuriken, which allowed him to only damage his opponent and not himself by staying out of the damage radius.
When Naruto first created the Rasen-Shuriken, the technique was only half-completed. It wasn't stable enough to be thrown, so Naruto has to use it as a melee attack like the standard Rasengan. However, the kickback of the jutsu is extremely dangerous.
The when the Rasen-Shuriken strikes, it dissolves into a veritable armada of microscopic wind blades that attack every cell in the opponent's body, severing the the victim's Chakra network. However, due to his close proximity to the point of impact, Naruto experiences similar aftereffects in the arm that wielded the technique. While the damage to his Chakra network wasn't nearly as severe, if overused, it would render Naruto permanently incapable of molding Chakra.
However, after Naruto learns to enter Sage Mode, he can use the Sage Chakra to stabilize the Jutsu's form, allowing it to be thrown like a real shuriken. This is also true for when Naruto gains control over the Nine Tails' Chakra.
As a long range technique, the Rasen-Shuriken no long poses any danger to Naruto.
Naruto is able to use the Rasenshuriken freely basically because he enters sage mode which allows him to throw the Rasenshuriken because of the nature energy and his own Charkra combined granting him the ability to throw it, in his base form he couldn't he had to attack Kakuzu up close like an regular rasengan, which damages both the user because he had to get up close, and the victim.