Astronomers understand big flaming balls of gases very well, and there is very high confidence that lone F3 main-sequence stars cannot have variations of more than 0.01 percent on any time scale shorter than millions of years. (Well, except for superflares, which these are not.) Further, the light curve dips are not modulated by the known stellar rotation. So the dips aren’t from the underlying star.
This forces us into thinking that they could only be due to something passing across the line of sight, blocking the background starlight (the technical term is “occulting”). After all, the PlanetHunters were explicitly looking for exactly this case, where some orbiting planet occults the star as it passes in front (the term for that is “transits”). But planet transits are always very small in amplitude (less than about one percent or so) and periodic. Whatever it is that is randomly passing in front of Tabby’s Star is not a planet.
The deepest dip dims the star by 21 percent, so whatever was covering up the starlight must have been covering at least 21 percent of the star’s area. This implies that the body, or cloud, or structure must have just under half the radius of the star—which makes it 70 percent the size of our Sun. Whatever it is, it might be solid (like a planet), gaseous (like a star or a gas cloud), or something like a cloud of dust.
The trouble is that any such idea requires that all of these things should be emitting large amounts of infrared light. It should be bright enough to be readily seen above the normal expected infrared light from the primary star. In astro-jargon, Tabby’s Star does not have any excess of infrared light.