*) At the top layer: The term in question has L=1 and S=1. Therefore
there are three values for the Lz component and three "independent"
values for the Sz component. (The values are not truly exactly
independent, but the LS coupling is small compared to the splitting
between terms.)
*) At the next layer: You can discover that L=1 and S=1 by counting
electrons and then applying Hund's rules.
*) At the next layer: Hund's rules are well founded in atomic physics.
The best explanation I've seen (indeed the only explanation I recall
seeing that made sense) is in Baym.
In particular, the explanation of Hund's rule #1 found at the (otherwise
generally very good) hyperphysics site: http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/hund.html
seems bogus to me, as if it were written by somebody who had once
seen the right answer, but didn't understand it, and is now trying
to reconstruct by rote something that "sounds right".