Internships at the Houston Zoo are designed to be educational experiences, but you should have an academic or professional background to some extent in the area that you want your internship. To be most competitive as an animal care intern, you should get experience working with animals in other capacities (animal shelters, wildlife rehabilitation clinics, or large animal work for example).
Going above and beyond the classroom setting will prove your passion for this field! Seek extra-curricular experiences that will showcase your ability to work on concepts related to the focus of your desired internship.
If your skill set does not lend itself to the internship you want, be sure to explain fully why you are the best candidate in your cover letter. Work ethic, the ability to get along with others, utilizing good judgement, and enthusiasm for conservation are valuable skills that are not career-specific.