How to identify branches of root locus from figure/plot?
The number of branches will be equal your number of poles. The different colored lines should indicate each branch (unless you're colorblind, the difficulty of interpretation might be challenging). 

When two poles collide with each other (departure), they will go off to try to meet with a zero (arrival). The trajectory to the pole would be a different subject but that part is called the "angle of departure". Once it arrives to a zero, that's called "angle of arrival". If there are no zeros around that pole, it could go off into infinity depending on your value \$K\$. Each of these departures is considered a path. 

(I could be missing something here as it's been a while since I've done a root locus.)