师 (traditional 師) originally meant an army or a great multitude — many people gathered. From 'leader of the many' it came to mean a master or teacher: the one others follow and learn from.
老师 (lǎoshī) — 'teacher' — is one of the most respected words in Chinese culture, rooted in a 2,500-year tradition of revering teachers since Confucius (himself called 万世师表, 'teacher of all generations'). 师 also forms 师傅 (a skilled master/craftsman, also a polite way to address drivers and workers) and 律师 (lawyer), 工程师 (engineer).
The one who leads the multitude — a MASTER, a TEACHER.
师 is the respectful suffix for skilled professionals: 律师 (lawyer), 工程师 (engineer), 厨师 (chef), 师傅 (master craftsman). To add 师 is to grant respect.