Edmund Gwenn, the chief scientist assigned to the case. He and his also scientist daughter actually postulate the incredible theory of what's happening just by examining the footprints. It is they who hatch the plan (pardon the pun) to eliminate the nest, investigate and discover two queens have escaped the New Mexico desert where the original nuclear testing was done in 1945, and convince the U.S. military of the urgency to find them and hunt them down. A 24-hour media monitoring effort ensues to look for any evidence of their location. One queen ends up settling onto a Navy ship which is eventually sunk, and the other heads to the underground storm drain system of Los Angeles, where they have to go in and rescue two children before they can torch the place.
Watching it now, I can really appreciate the quality of the screenplay, acting, and direction. The focus is on the story and characters more than the ants themselves, which are only shown when necessary to advance the plot elements, and with a good balance of action and drama. You also have to admire the fact that the ants were all life-size mechanically operated replicas. Given that, they don't look half bad. The Wikipedia write-up says it was nominated for its special effects and that it was the first "big bug" movie, bestowing it with at least some historic precedent.
Below is an original trailer. Don't be too put off by the dramatic music and opening titles - it was pretty standard for the time period. And its interesting that the beginning quote in this clip is actually uttered in the movie by the scientist in a private meeting, not proclaimed by a TV announcer. Yes, Hollywood marketing has been around a long time.