Google App Engine is a platform as a service (PaaS) that provides elasticity by allowing applications to scale up and down automatically based on usage. It achieves this through a stateless architecture where application servers can handle requests from any user, and the state is stored externally in a database. This allows instances to be shut down when not in use without losing session data. By moving state to a scalable NoSQL database like Bigtable, App Engine is able to achieve scalability to handle large traffic spikes or slashes.