Ultimately, tournament competition is not about winning medals. Competing is about performing in a new environment, under stress, against other members of the greater martial arts community. Through competing, you develop the skill to focus under pressure - you test yourself and grow from the experience.
There is a common saying that to compete is not to compete against others but against oneself.
What this means is that through the course of competition training you push yourself to higher limits, focusing on details of movement, developing better skills of sparring and more. This type of focused training pushes you to try harder, to work harder and to do better. These aspects of tournament training make you an all around better martial artist and this carries over to other aspects of your daily life. While competing is by no means mandatory as a martial student, it definitely has its rewards.