You are right that it was connected with a CAT 5 wire or Ethernet. He said it was faster than connections with an USB wire.
It makes sense that his "Network Attached Storage" is faster than a USB drive. USB was built for universality, not speed.
Initial uses of USB were primarily for devices such as keyboards and mice and has evolved into much more. When v2 was released, it became popular for hard drives and more, though still relatively slow.
Modern versions of USB are actually quite fast, and the latest version (type-c) is very fast. Compared to Gigabit Ethernet (the more common modern version of networking, though the slower 100BaseT Ethernet is still used in many locations as it requires Cat 5e or 6 cabling) the newer versions of USB are better for accessing storage. Keep in mind that ethernet, and the server hard drives, has overhead so you cannot simply compare theoretical speeds of the cable to others.
Network file servers may perform at 40-70 MB per second on a local network with gigabit Ethernet, which is significantly less than the transmission speed possible with gigabit Ethernet. USB 2, which has been available for many years, works at a maximum of ~60 MB per second. Compare this to USB 3.0 at 640 MB per second. Unfortunately, many devices (especially cheap products) just aren't designed to provide maximum speed so even a USB 3 device may not always perform at these speeds. The new USB 3.1 and Type-C versions are even faster.
And, most NAS devices or inexpensive hard drive sharing solutions (such as connecting a drive to a router) will almost certainly perform poorly when compared to simply attaching it locally via USB 3. USB 3 hard drives are actually quite fast from a practical perspective. In other words, your local drives will likely be faster than any network resource.
https://en.m.wikipedia.org/wiki/USB
https://en.m.wikipedia.org/wiki/Gigabit_Ethernet
A good review of Gigabit Ethernet here, though some tech changes in recent years have increased performance since this was written:
http://www.tomshardware.com/reviews/gigabit-ethernet-bandwidth,2321.html
Brief overview of USB 2 vs USB 3 practical performance:
http://www.pcworld.com/article/2360306/usb-3-0-speed-real-and-imagined.html
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