You need to have had a good amount of experience in publishing to become a good literary agent.
This can be acquired through internships, which are usually unpaid (although a lawsuit a few years ago has made many agencies wary of taking on interns now), or through working in other areas of publishing.
A degree which helps you write clear, perceptive prose is useful. So, and English degree, or a history or politics or economics degree.
My niece was offered a three-week internship in one of London's best literary agencies: she was at the time studying for an English Lit degree at QMC, which is a very well-respected course. She loved the work at the agency and was asked if she'd write a reader's report: she did, and they were amazed by how good it was, and how quickly she wrote it.
Her three weeks there stretched to three months, her whole summer holiday.
She spent her final year at university writing reports for the agency--paid work--and the week before she handed in her final dissertation, was asked if she'd go back as a temp. After a month she was given a full-time, permanent job. She was there for two or three more years, then was head-hunted by a larger agency.
She's not an agent: she works with rights and contracts. But she's still writing those reports, which are a big strength for her, and says she couldn't have done them if she hadn't learned so much about literature and essay-writing at university.