I want to stay in Chicago, why should I send my resume to recruiters in Los Angeles?

A recruiter in Los Angeles or New York or Dallas or Atlanta could be working on an assignment in Chicago. The recruiter will contact you if you are a match for the assignment, regardless of location. If you are location-restricted, state that in your cover letter. But even if you cannot relocate, becoming known to recruiters who work in your field is a good job-hunting strategy.