"These dips could have been caused by a planet orbiting the
binary system, but in 2013, a different group proved that
the third object must be a star. They did this by finding
variations in the orbital period of the binary, variations
caused by the changing light travel time as the binary
orbits a third star."
Translated: the binary star system is co-orbiting a third object, whose pull on the binary is so strong that it must be star-sized rather than planet-sized. How do they measure this pull? It's a Doppler effect -- not the familiar one, a shift in the frequency of waves (like sound) -- but a shift in the frequency of the binary star orbit! This orbit is a clock with a constant 0.258 day period between ticks. When the binary is moving towards earth, these clock ticks are "catching up" [0] with the starlight moving towards earth -- they're compressed together, so the binary looks like it's orbiting faster than it actually is! If you measure this Doppler shift, how it changes as the binary orbits the third star (first moving slightly faster towards earth, then away from it), you can infer the speed of its orbit around the third star -- and from that, the mass of the third star!
I was just reading about this recently: I was going to say (wrongly) this technique was used by in the 17th century to measure the speed of light, by measuring a Doppler shift of the orbital period of Jupiter's moon Io [1]. In fact Ole Rømer measured the phase of Io's orbit, rather than the frequency. You can track Io's orbit phase very exactly by timing the moment it falls behind Jupiter's horizon (a regular eclipse). With Io's orbit as a clock located at Jupiter (the eclipse moment as the "tick"), you can measure differences in light travel time from Jupiter to Earth as they move apart. This is a phase measurement, not frequency; and it depends on the Jupiter-Earth distance, not their relative speed.
(Disclaimer: this has nothing to do with special relativity, despite dealing with the speed of light! Delays in observing an event, due to light travel time, are not what relativity is. This is simpler stuff).
(Also, don't confuse this with doppler spectroscopy [2] -- measuring shifts in the frequency (color) of light, rather than the frequency of orbits. This is the "Doppler" that's used to discover exoplanets!)
I was just reading about this recently: I was going to say (wrongly) this technique was used by in the 17th century to measure the speed of light, by measuring a Doppler shift of the orbital period of Jupiter's moon Io [1]. In fact Ole Rømer measured the phase of Io's orbit, rather than the frequency. You can track Io's orbit phase very exactly by timing the moment it falls behind Jupiter's horizon (a regular eclipse). With Io's orbit as a clock located at Jupiter (the eclipse moment as the "tick"), you can measure differences in light travel time from Jupiter to Earth as they move apart. This is a phase measurement, not frequency; and it depends on the Jupiter-Earth distance, not their relative speed.
[0] Like this GIF: https://en.wikipedia.org/wiki/Doppler_effect#mediaviewer/Fil...
[1] https://en.wikipedia.org/wiki/Rømer's_determination_of_the_s...
(§8.2 says my confusion is a common one, so there!)