Why does a helium balloon fly?

Unfortunately your answer is wrong.
The weight force of the balloon must be smaller than the lifting force, so the balloon can take off.

Because of the fact that helium is lighter than air, the weight force of the helium balloon is smaller than the lifting force and so the balloon takes off.

Back to our example:

The balloon with a volume of 5 litres is affected by a lifting force of 0,064 Newton.
5 litres of helium have got a mass of about 0,9 g which is 0,0009 kg. Additionally we have to add the mass of the rubber skin of the balloon which is about 5g. Thus the balloon has got a total mass of 0,0059 kg and a weight force (F = m*g) of about 0,058 Newton. As you can see from our example, the lifting force (0,064 N) is greater than the weight force (0,058 N).

An air balloon is filled with air, as the name suggests. The weight force of the displaced volume of the air and the weight force of the air within the balloon is equal therefore. As a result, the lifting force and the weight force of the air within the balloon are equal. But here too, we have to add the mass of the rubber skin of the balloon to the mass of the air within the balloon, therefore the weight force of the balloon increases additionally. Finally the weight force of the balloon is greater than the lifting force. The air balloon falls to the ground.