6 months after graduation is typically when your loans will begin to require repayment. However, you can apply for deferral of payment based on financial hardship and other criteria. This allows you the loophole of not beginning to repay loans during residency if you can't or opt not to. Your loans will accrue interest during this time period though.
Often the interest rate on educational loans is lower than others such as car loans or mortgage payments that you might be making at this point in your life. Therefore it is frequently a wise financial decision to defer loans and repay them over a longer period of time, allowing you to put the cash on hand into other necessities or investments.
Long story short: No, you do not need to start repaying loans during residency.