I believe the discovery was made by orbiting satellite, but I'm not sure which one.
That is not the case.
Look at the author affiliation for the article to which you linked. The three authors of that paper were from the British Antarctic Survey. These scientists were part of a larger expedition to Antarctica. They pointed a cheap instrument (extremely cheap compared to a satellite-borne instrument, and rather cheap compared to the cost of sending scientists to Antarctica) called a Dobsonmeter at the sky and found rapidly declining ozone levels at the end of winter / early spring in 1984.
The ozone hole could have discovered by a pair of instruments on the Nimbus 7 satellite, the Solar Backscatter UltraViolet (SBUV) and Total Ozone Mapping Spectrometer (TOMS). That satellite was launched in 1978, but those instruments did not detect the ozone hole because the ground software that processed the data from those devices marked the relevant data as missing rather than as very low.
There were solid reasons for doing so. Remote sensing works best when the readings are corroborated by more direct techniques, in this case by rocketsondes and balloonsondes that can directly measure the constituents of the atmosphere as a function of altitude. (The suffix "sonde" is a fancy way of saying "probe", or more precisely, a sounding probe.)
Readings as low as those implied by SBUV/TOMS had never been observed by these direct techniques, or even by ground-based remote sensing devices such as a Dobsonmeter (at least not until 1984). There was no validation for those very low measurements. Moreover, SBUV/TOMS worked by measuring ultraviolet light that was backscattered by the atmosphere. This technique was deemed to be perhaps suspect when the Sun was very low on the horizon, which was exactly the case in late winter / early spring in polar regions.
The SBUV/TOMS ground processing software, which had to process lots and lots of readings and which had to run unattended, simply rejected readings that were either overly low or overly high and when the Sun was low on the horizon. I wrote the software that made those checks, per requirements made by the team scientists. Even though I left the Ozone Processing Team in early 1980, I received a number of phone calls from NASA in late 1984 / early 1985 because "your name is all over the software." Fortunately, it was easy to fix: Simply relax those constraints and rerun the software on the historical data. NASA soon found that the ozone hole went all the way back to 1978, when the satellite was launched.
How it works
Oversimplifying, the underlying science is the same reason the sky is blue, which is Rayleigh scattering. The sky is blue because the atmosphere scatters blue light much more than it scatters red light. Since Rayleigh scattering is inversely proportional to wavelength raised to the fourth power, the atmosphere scatters ultraviolet light even more strongly than it scatters blue light. While some of this light is scattered toward the surface of the Earth, some of it is scattered out into space. Looking at the frequency distribution of this backscattered light gives insight into how much ultraviolet light is being absorbed by ozone in the stratosphere, and this into how much ozone is in the stratosphere.
That was oversimplified. There are other types of scattering involved (e.g., Raman scattering and Mie scattering). Clouds reflect rather than scatter. Aerosols change the Rayleigh coefficient and other scattering coefficients. The angle at which sunlight hits the atmosphere has a significant effect. These and other effects are taken into account, and the algorithms have been tweaked repeatedly so as to better comport with direct measurements.