It is seriously difficult if you don't understand the meaning of the terms used.
Okay, so that resolves to "bandwidth" being the problem term. I can explain that:
Think of a simple AM radio signal. A carrier frequency (the frequency you tune the receiver to) is "amplitude modulated" (AM) by the transmitted content. This means you take the audio signal you want to transmit, and use it to modulate (ie vary) the amplitude (ie, if you like, the signal strength) of the carrier wave. Alright so far? So, if you looked at the actual radio signal on an oscilloscope (a device which displays a trace of the variations of a signal over a short period of time), you would see the rapid oscillations of the RF increasing and decreasing in overall height in accordance with the audio signal imposed on it.
The mathematics of Fourier, Shannon, Nyquist
et al demonstrate that modulating a carrier wave spreads out its frequency spectrum. A pure sine wave has a specific frequency. A modulated sine wave has a frequency spread roughly equivalent to the frequency of the modulation - in this case the audio signal being carried by the radio frequency carrier wave. If you, say, limit the range of frequencies in the audio signal to 0-5kHz, the carrier spectrum spreads 5kHz either side of the carrier frequency and is capable of causing interference to another AM radio signal if its transmission frequency is too close. That's why the radio stations broadcast at carrier frequencies 9kHz apart.
Thus we have the bandwidth occupied by a UK AM radio station: 9kHz (or a bit less actually, but good enough for illustration). But that's for just one station. The whole medium-wave AM radio band goes from about 500kHz to 1600kHz, a bandwidth of 1100kHz.
Light in an optical fibre is just the same, but its frequency is much higher and the range of frequencies that can be transmitted through glass fibre go from roughly 300,000,000,000,000 Hz to 1,000,000,000,000,000 Hz - that's a potential bandwidth of 7x10^14 Hz. However, the bandwidth actually being used depends on the lasers etc that launch the signal into the fibre and the detectors at the other end.
Shannon worked out the theoretical maximum payload (in the case of AM radio: the audio signal; in the case of fibre: the data it carries) that can be squeezed into any particular bandwidth, so the development of digital TV etc is about engineering new modulation schemes which get closer and closer to the Shannon limit - ie use the available bandwidth as efficiently as practically possible. Ordinary AM and FM are pretty inefficient in terms of information content per Hz of bandwidth.
However, that's just showing off, and I suspect post 388 was all the explanation required. The point of the article was that a workable fibre link has been accomplished over a length of fibre sufficient to get from London to New York without any repeaters in between. That means they can lay a single cable across the Atlantic without needing any optical amplifiers along the way - and amplifiers mean parts to go wrong, joints to leak sea water, and a means to power them (not easy, 2000km from a power supply). If you look at an apparently transparent pane of window glass edge on, it looks dark green - that's what it does to light coming in from the opposite edge and travelling through the width (not thickness) of the glass sheet. OK, so the glass used in optical fibre is much, much more transparent than window glass, but imagine the difficulties involved in recovering any photons at all after they have had to travel through nearly 6000km of it.
