Here's how it works (this is my opinion as a web developer, not verified details from Snowden leak):
1) Fancy user interface developed by Booz Allen Hamilton. Enter email address (good choice for a unique identifier, used as unique key in many databases).
2) Backend uses curl to send a request to NSA-certified web api on each service shown on the slide. This serves as a legally-binding FISA request either regarding a foreign agent - no court order required - or a domestic agent - secret FISA court order required (see http://www.npr.org/2013/06/13/191226106/fisa-court-appears-t...) and assumed to be fulfilled by the api.
3) Kick off NSA equivalent of gearman worker that checks contents of "dropbox"-like service from each company for updates.
4) Services (Google, Facebook, etc) automatically grant request without question as it is a legally binding FISA order. This saves them a ton of money and, hey, it's legal! They have some custom code that allows them to look up a user by their email address - almost guaranteed to be indexed in their database - join it to relevant data sets, and dump it to the "dropbox"-like system.
5) Fancy frontend shows progress bar, while skinny backend compresses retrieved data into zip file for easy download.
This is the most efficient, cost-effective way to do this without venturing into science fiction, ie storing a mirror of all the data which would be stupid on NSA's part. It still verifies our worst fears and answers the question as to how such a program can cost "only" $20 million per year as reported by the slides.
The trick is in the legal framework, not the technical details. This is why the FISA courts are secret.