Why do elements in a family have similar properties?

The answer to this question comes largely down to quantum mechanics. If you've noticed, the period table has a funny shape. That's because the elements are not only arranged in order of increasing atomic number (number of protons in the nucleus), but they are also arranged by electron configuration. Electrons are arranged in atoms in a certain way. First, they are arranged in shells, represented in the periodic table as horizontal rows, or periods. Elements in the same row have their outermost (or valence) electrons in the same shell, which means they have approximately the same energy. You can't fill in a shell until the lower shell is completely full, which makes the horizontal row structure of the periodic table work. Next, within each shell, as you fill in each shell, you place electrons into sub-shells, and this is where the family comes in. Basically, elements within the same column (family) but different rows (periods) have their outermost (valence) electrons in the same sub-shell of different shells. While this does not mean that the elements are the same by any means, it does mean they will generally share some characteristics because their valence electrons are in the same configurations and thus behave very similarly.

A more advanced explanation would have to not only include the myriad reasons why similar electron configurations lead to similar physical properties, but also include a treatment of hybridization (where sub-shells "merge"), etc.

Thanks for the question and the chance for a washed-up physics major to exercise his brain.