Most of Spider-Man: Homecoming is set years after New York was invaded by aliens in The Avengers. But ComingSoon.net reports that the first scene actually takes place directly in the aftermath of the climactic battle, showing how Adrian Toomes was set on the path to becoming The Vulture.
The Avengers didn't really deal with the clean-up and recovery effort set off by the hugely destructive Chitauri invasion - it mostly focused on post-battle shawarma cravings. Very early spoiler warning: In Homecoming, it turns out Toomes and his company have secured a tidy contract to help clean up the mess. But just as they're about to begin, they're stopped by the Department of Damage Control and sent away.
The feds don't want civilian crews cleaning up the Chitauri mess with all those high-powered alien weapons and anti-gravity vehicles laying around. Toomes doesn't really care about their concerns since they put him out of a job. As a middle finger to the man, Toomes spends the next few years pilfering all the Chitauri technology he can. By the time Spider-Man: Homecoming catches up to the present, his company has been run out of business and Toomes is pursuing a more lucrative career as a supervillain, like you do.
Then Homecoming jumps over to the events of Captain America: Civil War, where it shows how Peter Parker got enlisted to help out in the big airport fight and what kind of fun stuff he got up to while he was still in Europe. Then… back to school life.
“After seeing all this amazing, glamorous stuff,” executive producer Eric Carroll explained, “boom… hard cut to him back in his cold weather gear. Riding the subway to school. Crammed next to all the other students, riding public transportation to school and we’ll realize that he’s still waiting for his call from Tony.”
