When a planet with an atmosphere eclipses its star, light passes through its atmosphere and the molecules in the atmosphere will absorb specific frequencies of light.

See https://exoplanets.nasa.gov/discovery/how-we-find-and-charac...

I would guess spectral analysis of the light coming from the planet's star. Look at the spectrum when the planet is behind the start, look at the spectrum when the planet is in front of the star (in which case part of the light passes through the planet's atmosphere). The difference between the two indicates which frequencies are absorbed by the planet's atmosphere, from which its composition can be determined. Don't take my word for it though, I'm not an astrophysicist.

In any case it boggles my mind that we can do things like that for planets thousands of light-years away from us.

They use a technique where they do spectral analysis on the light reflected from the planet. Gasses absorb different wavelengths of light. So the light reflected will have holes at specific wavelengths, and scientists can determine which combination of gasses will result in those specific absorption lines.

I thought it was just light? Light bounces off of different molecules differently?