Disclaimer: I’m playing fast-and-loose with physics in the following and in my storybook for the sake of style, narrative convenience, the broad background of the readers, and my own ignorance: I’m a half-baked mathematician, not a physicist, so I’m also coming at this all from a very different angle than a physicist would.
To get a sense of the… “disagreements among friends” between mathematicians and physicists, consider this quote from Einstein:
Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.Albert Einstein (1949)
The Arecibo Observatory is a radio telescope — it receives and resolves EM (electromagnetic) waves at radio frequencies, which are those ranging from the mid kHz (thousands of oscillations per second) all the way into the upper GHz and beyond (gigahertz, billions of oscillations per second). Radio waves have lower frequencies than visible light, which ranges roughly from 430-750 THz (see here).
(Visible light, ionizing radiation, ultraviolet rays, microwaves, and radio are all just names for different frequencies of electromagnetic waves: at some level, they are all the same.)
Because the frequencies of radio waves are so much lower than those of visible light, their wavelengths — the amount of physical space a single repetition, or period, of the wave “takes up” — are much larger. What does this mean in practice? While something as small as our eyes or a camera lens can spatial and spectrally discriminate visible light, much, much larger “cameras” are needed to see radio with any kind of clarity. This is why Arecibo is so large.
(These days, from what I understand, single large-dish receivers like Arecibo have gone out of favor and have been replaced largely by complex mathematical interpolation among large arrays of spaced out smaller receiver disks. This has great scientific advantages, and has enabled a great number of breakthroughs, but lacks drama. And drama, of course, is our goal.)
The sheer scale of a telescope like Arecibo presents significant difficulties in, well, pointing it at the things one wants to see. To some extent, it can only see what’s “right in front of it.” Radio telescope astronomy is further complicated by the fact that the radio waves must pass through Earth’s atmosphere before arriving at the telescope, and the atmosphere is anything but transparent to those radio waves. Some are fully blocked, others attenuated; weak signals from far-off galaxies commingle, cancel, and interfere with television signals, military radar, atmospheric events and weather, and that local classic rock 97.3 FM station you wish would stop playing the same three Rush songs over and over.
Arecibo and radio astronomy’s list of achievements is long and fascinating; one example of many is that, in 1989, Arecibo produced the first ever radar image of an asteroid, the “potentially hazardous” (if you’re feeling like panicing about something tonight) 4769 Castalia. (See here.)
A radio telescope also detected the as-yet-unexplained “Wow!” signal in 1977, an exceptionally strong burst of radio signal lasting some 72 seconds at the “hydrogen line” — 1420.41 MHz, the emission frequency of hydrogen, the most abundant element in the universe. It came from somewhere over Sagittarius-way. Though it was simply a burst of radio energy — not any kind of encoded message — it remains the strongest candidate for an extraterrestrial broadcast ever observed, and no explanation for its origin has ever been widely accepted.
P.S.: Well, there have been some rather convincing explanations for the “Wow!” signal: this recent (2017) paper by Antonio Paris finds that a hydrogen cloud that follows around a particular pair of pesky comets fits the bill, and retroactive orbital calculations reveal that they were in the right place at the right time in 1977 to play that funky music, so to speak.