To know more about you and cross-question on your parents’ work/profession.
To assess your ability to learn from your environment.
To know how well are you acquainted with your family and the influence they have had on your personality, education, and up-bringing.
How to answer this question?
Do not boast about your father/mother in case he/she is in a senior position. At the same time, you should not look down upon your parents’ occupation. It is important that you be factual about the answer. E.g. (My father is a manager in TISCO, Jamshedpur. My mother is a home maker and my brother is doing his graduation from "…")
Mention the impact your family environment has had; the values your parents have tried to instill in you, the emphasis laid on personal growth and education. You may be quizzed about your parent's occupation and you should be in a position to answer those questions. If your father has worked with TISCO, then you should have an idea of the Steel Industry, the kind of responsibility he has, etc. If your father/mother is an IAS Officer, you must have an answer for why you are not taking the Civil Services Exam. Interact with your parents about their profession before appearing for the Interviews.
Don'ts
Don't describe intricate details.
Don't undersell your family.
Don't try to gain sympathy by stating hardships or failures.
Concentrate on the strengths.
In short, you need to showcase your family background in a positive manner and remain as factual as possible. At times, you might feel that you are understating the details but such an approach works best in this case. Ideally, you need to prove that you are a suitable candidate and excessively relying on your family background might count against you.