the food in the FL is way better than the snack mix and veggies in the AC. The FL is located inside the AC, so you can go back and forth. The AC has better views than the FL, the FL is smaller, more intimate and has a full self service bar, coffee bar, soft drinks, juices, and real food depending on time of day. The showers in the FL are some of AA's best.

When you check into the AC, you'll get a card to access the FL, which is behind the frosted glass sliding door upstairs just south (right) of the AC desk.

I see no reason to leave LAX to get food, but some people like to make a run for In-N Out Burger, which is near the airport.