Warning: spoilers ahead for the first two episodes of the "Crisis On Earth-X" crossover.
The "Crisis On Earth-X" crossover brought a whole bunch of very bad guys to Earth-1 to battle the heroes of Arrow, The Flash, Legends of Tomorrow, and Supergirl. On the world known as Earth-X, the Nazis won World War II back in the 1940s, and everything is generally terrible for the innocent of that place. We've known for a while that the heroes of the Arrow-verse would be facing their evil doubles and Nazi cohorts in the annual crossover, but it was never clear exactly why the Nazis would choose to invade Earth-1 of all places. After all, there are a lot of Earths with a lot of different heroes in the multiverse.
The Arrow leg of of the "Crisis On Earth-X" crossover revealed just why Earth-1 was the target of the Earth-X Nazi invasion, and it was not what any of us probably saw coming.
Having fully conquered their own Earth, the Nazis decided it was time to take the war into the multiverse and attempt to conquer others. Earth-1 was selected because one very important person in the evil regime on Earth-X needed the help of her double. Overgirl -- that is, evil Kara of Earth-X who just so happens to be married to evil Oliver -- is dying due to an overload of solar radiation in her bloodstream, which the good guys were able to discover thanks to Oliver's successful shot with a kryptonite arrow. Her heart is going to give out and she will die unless she gets a transplant from another Kryptonian. As there evidently aren't any spare Kryptonians who are just Overgirl's size on Earth-X, she had to go after Kara, who just so happened to be on Earth-1 to attend Barry and Iris' wedding.
Unfortunately for the good guys of Earth-1, the combined forces of Overgirl, Reverse Flash, and throngs of Nazi soldiers were too much for even the assorted heroes from four different superhero series. Dark Arrow wasn't even present, and the bad guys were still able to get the win over Oliver, Barry, Kara, Sara, and others. The Earth-X crew kept everybody alive to use as leverage against Kara so that Kara would allow Overgirl to harvest her heart and live on. The heroes (aside from Kara) were transported to a concentration camp on Earth-X.
Of course, the Nazis do also want to take over Earth-1, so it's imperative that the heroes make it back home sooner rather than later. Hopefully they can escape quickly enough to get word to Kara so that she refuses to cooperate. Even weakened as she may be by the effects of red sun generated by the prism stolen on Earth-1 (and/or the kryptonite courtesy of Metallo), Kara is still a powerful fighter, and she'll certainly be motivated to fight back.
I'm wondering if perhaps the entire reason the Earth-X crew invaded Earth-1 rather than Kara's Earth-38 is that Earth-38 is the home of heroes like Superman and Martian Manhunter in addition to Supergirl. Formidable as the forces of Earth-1 may be with the likes of Oliver and Barry and the rest, the Nazis would probably have a lot more trouble with two Kryptonians and a Green Martian that they would with the active heroes of Earth-1. We'll have to wait and see.