To understand how peculiar, let's discuss mission parameters. The mission is to observe stars for signs of planets orbiting them. If a planet crosses in front of the star it will occlude the light. A planet the size of Earth would only reduce the light by a tiny bit, much less than one percent. A large planet like Jupiter might bring it up to a full percentage point. And this will happen on a regular basis as the planet orbits its star.
The light dips for Tabby's Star are 15% and 22% approximately 750 days apart. Is it the same object eclipsing the light, or is it two separate objects spaced equally apart? I suggest it is one object. Why? For one, it is many times more unlikely for there to be two light blocking masses diametrically opposed in the same orbit. Two, one object makes sense if it is the result of intelligent tampering, and I think it is. The rotation time tells us where an object is in relation to its star. If we know the period and the mass of the star, we can calculate the height of the orbit. We know the period is approximately 750 days or 1500 days. The mass of the star is about 1.4 times the Sun. If the artifact circles Tabby's Star every 750 days the orbit is comparable to Mars if it were in the Solar System. If it is two objects equally spaced in the same orbit taking1500 days, they are much further out. They are not in the "Goldilocks zone" and much less likely for life to form or to be out there doing engineering projects on a grand scale. So 750 days would put this object in a more likely orbit for life to form and, eventually, become intelligent. An orbit comparable to Mars is still too cold to be considered in the Goldilocks zone, but perhaps these intelligent beings are from a planet that is in the Goldilocks zone, and they are building at the 750 day orbit because the material they need for the project is already there or near there.
Why the difference in the dip in light? My suggestion is an artifact used by an alien civilization to construct their partial version of a Dyson swarm. A Dyson swarm is a net of habitats and energy collectors positioned in a loose sphere around a star to take advantage of its radiation output. A major portion of this artifact would be made up of mirrors that concentrate sunlight onto other material in the solar system to move it or melt it in the process of construction. Therefore, you are going to see different percentages of starlight dip every 750 days because they are going through phases of construction. Mirrors have to adjusted, habitats moved, etc.
What we are watching took place about the time the Roman Empire started to crumble. Think how fast our technology is going now, and we aren't even building giant suncatchers in space. That also means they don't know we are here yet because the Romans weren't broadcasting the Olympic games or I Love Lucy. It won't be until these broadcast signals reach them, or they see changes in the atmosphere due to the Industrial Revolution that we will give ourselves away. It will be the year 4840 A.D. before they could even respond to that information.
That this thing is only 1500 light years away is an amazing fact. If it is not some oddity of coincidence then that means there could be dozens, possibly hundreds, of such constructs in our galaxy alone. That giant chasm of time and distance also points out the localization and loneliness of an intelligent race.
The Earth came to the habitability party early. It has already suffered five mass extinctions. This world or civilization circling Tabby's Star may never have suffered any. Scientists believe that 92% of Earth-like planets haven't been formed yet so this could mean the herald of a boom in intelligent life taking control of it's environment.
If it is a civilization building an artifact we should watch them closely. They have obviously gotten past the need to burn things for energy. Perhaps we can pick up a few tips.