Martial arts were developed for self-defense during the dynasties of old in Japan, China, etc.. because the people were not allowed to have any kinds of weapons so they could not rebel against their rulers. So they figured out how to fight with their bare hands and with common farming tools (the nunchaku was realy a tool for mashing grain, the sai was a tool for plantingm, etc..).
I took lessons for a couple years in Tang Soo Do (similar to Tae Kwan Do) and American Goju Karate. The bowing is simply an old-fashioned, eastern (asian) way showing of respect to your teacher(s). We did practice meditation but it was only to learn how to concentrate and focus our minds so our thoughts are not about what stressed us out that day, etc.. it was not anything religious.