It's all about which atom has the higher atomic number.
If there's a tie (which is often the case with carbons), you go out to the next atom.

So in this case, you have:
a) a carbon bonded to three chlorines -> C (Cl, Cl, Cl)
b) a carbon bonded to a bromine, hydrogen, and another carbon -> C (Br, C, H)

So it's C (Cl , Cl, Cl) vs. C ( Br, C, H). (Notice that I put the atoms bonded to the carbon in descending order of atomic number, to make the comparison easier).

So which one is higher priority?

The two carbons are both carbons (duh!) so they tie.

But for a), the next highest atom is a chlorine (Z = 17).
For b), the next highest atom is a bromine (Z = 35). So C ( Br, C, H) is higher priority than C (Cl , Cl, Cl).

Notice that it doesn't matter that a) has three chlorines- after the first comparison, a) already loses!

There's a more detailed R/S configuration problem at the very end of this article:
http://www.mendelset.com/articles/688/molecular-models-and-stereochemist...