You're misinterpreting the map, it's showing two levels, not one. The lower part of the map is the lower level. The gold lines up with the upper, as does the grey road (two level road).

The ATS is outside immigration and customs. You'll come off the plane, go down the hall to immigration, then grab your bags, go through customs. Right after customs, there are baggage check desks where you can give your bags to AA (I think they have to be already tagged all the way through. if they aren't you may need to take them with you on the ATS). After the baggage desks, you exit into the main lobby/greeting area. Follow the signs to the ATS (up one level of escalators) and board the train to terminal 3.

Trains go in two directions, one is to the other terminals (1/2/3), the other to parking. Get the right one so you don't have to take the trek to parking and back Terminal 3 is the next stop.