Many places that offer internships will require you to be a student, and to have proof that you're earning some sort of college credit for your internship. That way, their compensation to you is class credit rather than a paycheck.
Others may accept recent college grads as interns. Even then, the pay will be very low -- usually just a little more than minimum wage.
You'll need something on your resume that shows your enthusiasm for the written word. I'm sure agencies get tons of applications from standard-issue English majors. Try to do something else that'll help you stand out -- work for a college literary magazine or read the slush pile for a literary journal. Intern for a university press. Write columns for the campus newspaper. Submit your papers for undergrad research conferences. Win a writing award from the English department. Have an interesting combination of majors and minors that'll bring an extra layer of experience to the agency -- say, an English / marketing double-major with a minor in psychology.
And if you're called in for an interview or do an interview over the phone, wow the agent with an absolutely exhaustive knowledge of the industry. Learn all the literary agency terminology. Learn the names of the major publishing houses and some of their most well-known imprints. Learn the names of the editors who work at those imprints. Read the NYT bestseller list every week and know exactly who's on it. For the agency you're interviewing with, learn the names of their clients and the books the agency represents.
I think it'd be tons of fun to intern at a literary agency. Good luck with it!