In traditional Japanese martial arts this term refers to a stage in the training of a martial artist in which he has completed his studies at his original school under his original teachers and he moves out in to the world to test himself and his understanding against outsiders. A shugyosha would travel from dojo to dojo requesting matches in order to test his technique. It is my understanding that sometimes these visits were structured by his teacher, going to recommended schools to train under teachers his own teacher recommended.Other times it was strictly done on his own. If he encountered a school in which the students or a teacher was able to best him, he might ask to stay and train with them. If he won he moved on looking for those who can teach him something.
In the old days there could be serious consequences as the matches were quite dangerous and often a school's survival could hinge on not being seen to be inferior to another style. The stiry of Musashi's shugyosha period are indicicative. This is the stage at which the student can becomes a master, if he survives. It is where he learns to apply his knowledge in the real world against styles not his own. Very dabgerous, very severe.