Hello @caitlynk13 ...
Are you using a Free for Teachers account, or does your school have a paid Canvas environment? The way it works for us is that people are first created as "users" in Canvas. Then, they are enrolled in courses with either a "Teacher" or "Student" role. If an instructor or a student is added to a course prior to being added first as a "user" within the Canvas environment, then we would see an e-mail address listed in the "People" area of the course...as you've described in your question. But, because people are already in Canvas as "users", the instructor/student names appear correctly in the "People" page of the course rather than displaying their e-mail addresses. You might want to check with your school's local Canvas administrator about how to get this corrected within your course.
I hope this will help a bit...thanks! Take care, stay safe, and be well.