Most people transfer their old balance to a new credit card, then stop using the old card and start using the new. Your payments to the new card go to pay off the oldest debt, ie: your balance transfer. So you're paying off the 0% interest portion of the money, while being charged whatever interest rate for the new purchases.
The only time you should transfer money to a card with 0% balance transfers is if you plan on never using the new card and taking the time available to pay off the balance (ie: 6 months interest free on balance transfers, so pay it off before that 6 months is up)