It means that the person you are is a direct result of how you grew up. The "environment" you were brought up in influenced the decisions you make now. Your environment makes you who you are.
So for example, product of the environment implies that if one grows up in a home full of domestic violence, one of the children in that home may become violent as well when they have a family of their own. If someone grows up in the hood, that person may become a gangster. If somebody grows up in a descent home that is filled with good resources (computer, good books, television, radio, etc.), then they are more likely to become successful. If somebody grows up in a Christian home, they will most likely become a Christian. If somebody grows up in the Middle East or in a home with Muslim parents, they are more likely to be Muslim.