In order to store, analyze and interpret the reams of data flowing around the internet, the National Security Agency first has to capture it, by, for example, exploiting security holes or using custom radio chips to access computers that aren't hooked up to the internet. When all else fails, the servers and cables and data centers that make up the physical stuff of the internet give the agency a way in. Even if internet traffic isn't destined for or coming from the U.S., much of it still passes through U.S. fiber-optic hubs.
The physical organization of the internet makes the U.S. “a uniquely advantageous place to put listening devices,” says Christopher Mims for Quartz.
Since at least 2001, the US government has had access to all the internet communication passing through at least one and possibly more fiber-optic hubs in the US. That would give the NSA access to a vast amount of data without “direct access” to company servers.
The rest of this world doesn't really like this. According to Wired UK, Brazil and the European Union are getting set to drop $185 million to lay new fiber-optic cables across the Atlantic, in order to boost internet connectivity while simultaneously cutting the U.S. out of the loop.
At the moment, there are 600 communications connections between Europe and Brazil and there is only one that connects directly, without going through the US. That one only carries voice data.
… in the wake of Edward Snowden's revelations of mass surveillance, both Brazil and Europe have felt uncomfortable with the level of control the US has over communications infrastructure.
A single cable won't be able to supplant the dozens of cables originating from the U.S., but the partnership is a sign that foreign governments are not fond of what the NSA is doing, and are responding accordingly.