When watching this video, you should keep in mind:

The Lattice1 holds the keys and signs transactions. The mobile wallet simply requests signatures and broadcasts the results as transactions.

This video demonstrates manual signature requests, i.e. ones requiring the user to authorize on the Lattice1. In the future, the Lattice1 will also support automated signature requests based on permissions. For example, a user could set his Lattice1 to authorize ether transfers of up to 1 ETH per day, as requested by his Grid+Pay mobile wallet.

More demos will be released as additional functionality is built into the Lattice1 and pay app.