A road or path that branches off from another one starts from it and goes in a slightly different direction. If you branch off somewhere, you change the direction in which you are going.
After a few miles, a small road branched off to the right. [VERB PARTICLE preposition/adverb]
She branched off down the earth track. [VERB PARTICLE preposition/adverb, VERB PARTICLE noun]
Also
VERB PARTICLE]