The app basically uses Bluetooth Low Energy for the proximity measurements: The app generates temporary IDs, so if two smartphones with the app come close, they exchange their IDs and save the IDs of their respective partner locally and encrypted. If a person is tested positivive for the novel coronavirus, the doctor can ask the person to upload their list of contacts to the Pepp-PT server. The app can then compare its list with the list on the server and notify the user if they had contact with person who has COVID-19.
Source for the summary (before I found the site at the top): https://www.spiegel.de/netzwelt/apps/corona-warn-app-fuer-eu...