Get rid of the moon.
EDIT: As <PERSON> points out in the comments, this very much looks like the novel SevenEves by <PERSON>. Never heard of it, but it looks like an interesting read if you're interested in my idea.
Or have some aliens get rid of it. Perhaps they needed the materials present in it, and deemed its removal "compliant" with their code of conduct, which mainly dictates they can not directly cause the loss of other species' life.
-What about indirect loss of life - asked the captain.
-Nothing on the books, ma'am- replied her lieutenant.
Yes, shady individuals.
More on topic, the moon, or rather its more or less sudden disappearance, could have devastating results on our planet, as devastating as your feel comfortable handwaving them.
The moon gives our planet axial stability
The moon is what scientist believe made our Earth have a tilted spin (whith respects to our translational plane around the sun). They also strongly believe that it is what is keeping that tilt. If it were to disappear, that tilt could become much more erratic and cause much more sudden weather changes, and stronger ones.
Just imagine we go from 23º to, say, 50º, in the span of a couple centuries. The planet's cold and hot areas would change dramatically, and seasonally, which would make winds a nightmare.
The moon is (almost) the only thing that keeps our tides going
And along with their almost complete stop, ocean currents would greatly diminish (EDIT) tidal currents would greatly diminish, which is one of the primary factors in nutrient distribution for the ocean's ecosystem. They wouldn't stop, but it would be significant enough to cause massive loss of life in the oceans. This would be pretty much immediate, and would cause severe damage to the economy, particularly in already empoverished places, which would cause major political and economical unrest around the globe.
Increased volcano activity
Maybe not that much. Maybe a lot. Maybe just the oposite, less volcano activity. This last one is my favorite, because it would mean that presure would be building up under the Earth's crust. 188
The thing with the moon is that it doesn't just move water, it also moves our tectonic plates. No movement (or far less) would make them stop releasing the presure they build up, eventually resulting in one biiiiig BOOM (exaggerated for dramatic purposes).
Heavy moonrock shower
This one depends on the need of the alien race. Maybe they don't need the entire Moon, just about 50% of it. The other 50% is left there as a cloud of rocks, dust and other debree. It would rain down as a fire shower for centuries, potentially making the surface of the planet too risky to live on - hence the bunkers. This and evrything else I already mentioned.
It would affect everyone's psyche
Last but not least, there would be major unrest due to the uncertainty that watching your moon disappear would create, and the sense of defenselessness against a much more advanced bully race that just takes what they need and leave you to die. Morale would be just on the floors, and I would assume many people would start acting very irrationally, to the point that governments wouldn't be able to control populations any more, plunging the entire Earth into an environment more tipical of a few thousand years ago.
Final thoughts
Basically, it would trully be hell, and I don't expect much to survive. Though I think there's a good chance that humanity lives through it, and the Earth would eventually find a new balance. But I would assume it would take thousands of years.
Unless another, now benevolent alien race takes pitty on the Earth and puts a new Moon there. Or the same race judges those shady individuals and condemns them to restore it. Or not, and make everyone suffer. Up to you :) 258