If your brother says you broke his guitar and you say you weren't anywhere near it, your parents may ask your sister for an independent opinion, as she has no reason to lie. An independent person is free from the control or influence of other people.
The root word, depend, originally meant "to hang down." We use the word hang in a similar way. If the decision to build a new stadium hangs on funding, the decision won't be made till the money comes in. To be independent means that you don't depend on your parents to pay your bills. Or, for a three year old, it may mean going to the bathroom "all by yourself." It's all relative.