When the kids turn nine they are given bicycles. What does that signify for them?

This signifies that they are old enough to be, in some ways, responsible people.  This is because the bikes give them some amount of freedom to get places more easily.  The bikes also expose them to some amount of danger since they can fall off the bikes or crash them and get hurt.

I suppose you could think of it as sort of the equivalent of getting a drivers license -- it signifies that you are at least bit more grown up than the people who are too young to have them.