You answer with why you're interested in the company.
Sure, the recruiter reached out to you, but this was a scheduled call. You had some time to prepare. You also mentioned that the company is well known, so, even without research, you should know at least something about why you would want to work there.
- The company has a great reputation as an employer
- The company provides products you regularly use and enjoy
- They solve interesting problems
- You believe in the company mission and values
- You know people who work there and they speak highly about it
I would expect question 2 from the recruiter to start to dig into the kinds of work you like to do. This isn't necessarily about a particular role, but more of a "We have positions available. sevensevens' LinkedIn profile seems like a match. Does his experience and interests match any of our available reqs? Does he seem like Company material?"
As others mentioned in comments and answers, it would be good interview judo to give a reason or two and follow up with, "However, you're the expert on Company, why do you think I should want to work there?"
(I started to put this as a comment, but decided to include it in my answer)
You didn't describe a cold call, but a scheduled introductory interview. The LinkedIn message, email, call, etc to see if you were interested was the cold call. You are now a warm lead (you agreed to a 30 minute meeting). Now they are attempting to qualify the lead (in sales terminology) to understand if you're a good fit. That may be questions about your past experience, your future goals, and your interest in/opinion of the company. If they like you, they would tell you about the opportunity they think matches.
A cold call would be something like
This is Chris G, I'm a recruiter with Google, do you have 30 minutes to discuss an exciting opportunity? Great. Why do you want to work for Google?
and that would be asinine