Battle Any Member is the place where you need to go if you want to fight other, real players.
There are 2 methods: Live Battle and Offline Battle, although only Offline Battle is available at the moment.
In this mode you can type in the username of any account on the game to battle against that trainer's current Pokémon team. It is easy to find usernames, as you can open the "Members" tap on the left to see a list of all users as well as the top 100 users & Pokémon. There are also some accounts, such as "Training", that have been set up specifically for this mode.
It is possible to battle yourself in this mode by typing in your own username, which leads for an interesting mirror match.