The Genesect's home is the mountain range/snowy area where they originally start rampaging. New Tork City is nearby to the mountain range and the buildings remind them of the pillars of their original home which is why they start attacking, because they want to turn the city back into their homeland.
Mewtwo runs into them and finds out about their backstory, and how they were engineered by Team Plasma. Mewtwo tries to show them the way to cope with modern life but Extremespeed Genesect just won't take his s*** and go on with the original plan.
Ash and the others obviously get mixed into the conflict and it's basically Everyone VS Genesect Army until Douse Genesect ends up getting hurt/killed in the conflict and the Extremespeed Genesect learns the errors of his ways. Ash and Mewtwo go on with their plan and take them to a paradise where they can call home.
And that is basically my take on the entire movie.