Why does 'send letters' mean "apply for a job"?
In the United States, some companies will require for you to include a "cover letter" in your resume. A "cover letter" is a single-page letter written to the person or organization offering the job you're applying for. A cover letter is meant to be a summary of your resume, and usually includes a brief introduction about yourself, and encourages the reader to to read you resume. So when Andy said she'd "sent letters out everywhere", she meant that she'd sent out her cover letter, thus her resume, to many different companies.