How to reply to an academic interview invitation?

If they call you and you answer, be excited and happy and keep the conversation really short. The person who called you does not really want to talk to you, they just want to tell you that you got an interview and possibly to dump some information on you. Your side of the conversation could be:

This is so great and I am really excited to come for the interview. I have to run to a meeting/teach (doesn't matter if it is true or not) can we set up a time to talk in more detail later today or tomorrow?

At that point the person calling will either setup a time or tell you they will email you the information they need to tell you.

If they leave a voice mail, you can either call them back or send them an email, unless they tell you to do one or the other. If they email you, you should probably only reply with an email, unless they tell you to call.

In the follow up conversation/email there are things you need to know, which search committees often forget about. You should of course express how excited you are to come visit and tell them about your work and learn more about them and their cool work, but you do not need to directly sell yourself at this point (or really during the interview itself). What you do need to do is:

1. Setup a date for the visit. You should tell them when you are available.
2. Alert them to anyone you want to meet during your visit.
3. Find out how travel is paid for and arranged.
4. What the interview consists of and who will be at those events.
5. If they need a title and/or abstract, and if so when, for the job talk.

The reason you need to know what the interview consists of is that a campus visit may include one-on-one meetings/interviews, a panel interview, a research seminar on your past work, a teaching demo, and a "chalk" talk on your future work. If they want you to teach an hour lecture of an actual calculus 101 class, you really need to know that beforehand. Alternatively, they might want you to talk through a syllabus of a graduate class you would teach. A 10 minute research talk aimed at the search committee is very different than an hour long seminar with undergraduates present.