He was bitten by a (radioactive?) spider

...we just don't see it onscreen,

This Q&A on Science Fiction & Fantasy covers it fully but the reasoning for not including it in the movie was..

"We never talked about showing it, because the MO was always to not show anything we've seen before," Watts said. "If anything ever felt slightly familiar, to try to figure out if there's a way to avoid it or tackle it in a different way. But that was something that definitely felt like we've seen that before."

[...]

By now, a third Uncle Ben death in cinema in less than 20 years would be redundant. "That's why," Feige said, regarding the choice to skip the scene in Spider-Man: Homecoming. "You didn't really have to ask," Pascal added.

_{Comicbook, Uncle Ben And The Spider Bite Were Never Considered For Spider-Man: Homecoming}

One of the few times a studio has done something right in my opinion.

Spiderman’s origins are part of society's pop culture. Everybody knows his origin. It's been shown a thousand times.

Though one could argue that very young generations might not know, thus a scene should be warranted.

Homecoming completely ignores it and just goes straight into telling their story and delegates his origin to a single line while talking to Ned.

Peter: The spider’s dead, Ned.

Which is not only a nice quick thing to get over the scene, it has a nice symbolism that the spider and his origin doesn’t matter at all.