Why does this E2 reaction occur on the less substituted carbon?
E2 elimination is stereospecific for anti elimination. In both variants of compound B that you have drawn, the OTs and the proton on the tertiary centre are syn to each other, meaning that the elimination is disfavoured. 

For this reason, the NaOMe deprotonates at the secondary centre, generating the products as drawn, despite the fact that this goes against the Zeitsev rule which favours the most substituted alkene.