This article seems to make the most sense. As a hypothetical example, Google might have been manually fulfilling thousands of FISA requests. It sucked up time, took engineering resources, was error-prone, and lacked the legal paper trail to track such requests. In addition, the government wanted something faster, that used fewer agents to submit/collect data, and could more easily get updates.

So one day, an NSA agent negotiates with Google to build fisarequest.supersecret.google.com. NSA agents can directly upload FISA request documents, as easily as submitting expense receipts. In turn, Google's legal department can view each request and decide to comply with the click of a mouse, as easily as approving an expense report. If authorization is granted, the NSA can now view the emails of the requested user, and new emails can simply be viewed with a refresh of the browser. As Schmidt and Drummond say, Google can decline a request if it's improper or overly broad, and ask for additional information -- again, as easily as a manager looking at a questionable minibar expense on a trip report.

Said NSA agent does this with multiple companies, internally brands this as PRISM, puts together a Powerpoint deck, and declares victory. It's "direct access" since it's coming straight from Google and on a Google server, and it's "real-time" in that a request can be authorized quickly, and there are no more .zip files with data dumps involved.

Google has never actually heard of PRISM, and only knows that they built a tool to make it easier to do what they were already doing with legal FISA requests. To them, the NSA doesn't have "direct access," which is a loaded term for unfettered superuser access.

The entire program costs only $20mm because the government now requires only a few agents to submit requests and collect data. The cost of building the tools is borne by the companies, who see it as a cheaper way to comply with an existing legal obligation.