I will try to keep it a simple and short answer.
Is it possible to explain in simple terms what makes these waves happen?
The combination of solar activity (solar wind), Mars atmosphere and what is left of Mars magnetosphere.
Is a similar or at least related phenomenon detectable in Earth's atmosphere?
Yes actually there is one called Aurora Borealis or ...
The Martian nightglow comes from molecules in the atmosphere that react with solar radiation, carry it over to the night side, and give off the energy they gained as ultraviolet light as they free-form stable compounds. From https://www.nasa.gov/press-release/goddard/2020/mars-nightglow:
The diagram explains the cause of Mars’ glowing nightside atmosphere. ...