If you want to tap a classical signal traveling down a fiber, you want to turn that fiber into a beam splitter. A beam splitter probabilistically reroutes photons from the data stream down a different fiber. Since a single bit traveling down a fiber optic cable is represented by millions of photons in an identical optical state and fiber-optic lines are lossy, a few won't be missed. A 99/1 splitter (i.e. 99% one way, 1% the other) would be pretty hard to detect in a fiber optic line that is subject to fluctuating loss caused by bending, temperature change, etc..
The trick, according to this article, is to avoid cutting the fiber and splicing in a beam splitter. This is because, immediately after the fiber goes dark, the company running it will want to know where the fault is and hook up an OTDR to the line. In the simplest terms possible, this is a device that sends light down the fiber and looks at what gets reflected back. Any break in the fiber will reflect light back the way it came. Time how long the round-trip takes and you can pinpoint where the break is and send out a repair crew. If the operators are fast this can take mere seconds to do. Naturally, the NSA doesn't want repair crews finding their tap-in points!
So how do you turn a piece of fiber into a beam-splitter without cutting and splicing? Well, that's surprisingly easy. One way to create a fiber beam splitter is to take two fibers and fusion-combine them. Basically, you put them right next to each other and heat them up so that they melt together just a little bit. If you did this to a line in service it would never have to be cut. This is dead easy to do... to one fiber... in a lab. Doing it to a bundle of fibers in the deep sea is challenging! However, it's certainly feasible. Given that the NSA's budget dwarfs the entire scientific research budget of many nations, it's reasonable to expect they could get pretty good at it given a couple of decades.
They probably aren't doing this. I've done a little in free-space optics, but fiber really isn't in my bailiwick. There are probably other, much easier ways. However, based on a few minutes of thought I'm convinced the claims of the article are beyond merely feasible. I'd be shocked if the NSA, and every other nation out there with subs, doesn't have the same capability.
The real trick, of course, is getting the data back home (and processing it). The article suggests that the Jimmy Carter has the ability to lay fiber lines, presumably to take their purloined bits back to the nearest NSA processing center. The amplifiers spoken of are unlikely to be used to boost the signal that continues on the original fiber back to the people being spied upon, simply because a good method of tapping in should be indistinguishable from fiber loss, plus components like amplifiers spliced into the line are detectable! No, those amplifiers are likely there to boost the signal going down the NSA's pirate lines. What I don't understand is how the NSA keeps their lines from being detected. Wouldn't, for example, Russia patrol it's fiber back-bones (while tapping other nations fiber) and cut the NSA's lines? There may very well be an underwater struggle going on day-to-day that we're entirely unaware of!