As it has been said, rāmen is a world unto itself. There are so many different types, textures of noodles, which also vary in thickness and length. Additionally there are many different types of broth, which may or may not have seasoning, a hint of wasabi, green onions, or any other ingredients you can imagine.
I am only a student in this brave new world. However, I know enough to grasp the beauty of the kaedama system. The scenario is simple: you finish your noodles but you're still hungry. Rather than drinking the broth and calling it quits, you shout 「Kaedama!」 and all of sudden you have another full serving of noodles at a fraction of the price of the meal. For instance, a regular order of tsukemen (つけ麺) costs me 750円. A kaedama only costs me 110円 on top of that. You can even order a hankeadama (半かえだま) at some places, which is just a half-serving of noodles.
Interestingly most westerners consider rāmen a noodle dish whereas in Japan it's always considered a soup with noodles in it. Maybe that's because what's in the broth is often the tastiest.
There's no "right" way to eat the dish, but most agree the tastiest is to use the spoon to collect some broth and to dip the noodles into this broth, bringing both closer to you so you are not slurping noodles from such a great distance. To eat this dish with confidence is to slurp any noodles you grab with chopsticks all in one go. Nobody cares how noisy you are. Slurping like this also cools down the noodles so you don't burn your tongue if the broth is still scalding hot.