We just did that today in class. It's an E2, which means that it has to happen in one step and the leaving group and the leaving H have to be antiperiplanar. When you do the newman projections for the case where the most substituted and the least substituted C get the double bond, you should see that E2 happening on the more substituted side requires the leaving group and the rest of the alkyl chain to be in gauche configuration. Since both the leaving group N(CH3)3 and the alkyl chain are really bulky, that does not happen and E2 proceeds at the less substituted C.
BTW: The CH3I in the first step reacts with NH2 and results in N(CH3)3. I don't remember what the deal with the AgO2/H2O/heat was but I think AgO2/H2O lead to primary alcohol and then the heat made the elimination happen. But that would imply OH leaving too, so it does not sound right. Notes from class are not posted yet, so if you want the details, I can come back in a day or two.