The descendants are usually very much like the ancestors. Children grow up to be similar to their parents and people who know the parents will be able to see the resemblances.
The proverb is usually used in a negative way, as when an undesirable characteristic of the parent is noticed to be passed on the child. It is similar to the other proverbs: Like father, like son, Like mother, like daughter, and Chip off the old block.
He’s an moron, just like his father—the apple never falls far from the tree.