You can only access your Apple ID through iTunes on a Mac or Windows PC.
Im not sure of the exact reason but Apple require you to have either a Credit Card of iTunes Gift card linked to your Apple ID.
I have been so low on my Credit Card at one stage with only a couple of ¢ in my account, and even though i have the credit card linked to my iTunes Account, it wouldnt let me download a Free App.So in short you will have to ask a friend/family if you can link there credit card to your iTunes account, OR buy a $20 iTunes gift Card and redeem it, which you can do on your iPhone to download your free App.
As i said im not sure exactly why Apple do this, but they do. It could be to do with a age limitation on iTunes. Needing to be old enough to have a credit card, gives you the right age to have a iTunes Account.
Just doing some reading and minimum age is 13yrs old, BUT you have to have a parent or Gaurdian manage your account. So up until your 18 you use a parent or gaurdians CC when as once you turn 18 you can use your own.
HTH