It is important to remember that the Book of Proverbs is a book that contains a number of very short, pity, earthy sayings that reflect truth and also point towards the way that we should live our life in a number of different areas. The proverb that you have picked out clearly relates to the topic of how to be parent and raise your child. What the proverb recommends, is that we work hard as parents to train a child up "in the way he should go," making sure he knows right from wrong and the kind of values and principles that should govern his or her life. If we do a good job as parents in this, then that child will have that solid basis of values and principals for the rest of their life. If you like, the proverb uses a kind of metaphor to describe this process. If we show a child a safe path to follow and teach them just to remain on that path rather than go anywhere else, then when they are an adult they will remain on that path too, and not seek to find another path. This proverb thus talks about the responsibility of parents and how they should raise their children to form godly adults.