The importance of the plaid as it is known in North America became evident when Jamie and Murtagh were in Ardsmuir Prison. At that time, Scots were not allowed to wear their tartans, speak Gaelic or own weapons such as guns. This is represented by the clothing that the prisoners are wearing and how they are allowed to hunt while at the prison.
In Outlander, season 1, episode 1 “Sassenach”, we see an apparition of a Highlander who is looking intently at Claire through the window of Mrs. Baird’s Bed & Breakfast in Inverness while she is brushing her hair.
Frank comes upon the scene and notices the Highlander watching Claire. As Frank approaches the Highlander to ask if he can help him with something, the Highlander brushes past him and vanishes into thin air. We later surmise that this was an apparition of Jamie who was staring at Claire, but how, or why, did he come to be there?