Start off with "Hello, this is (your name), who is speaking?" and take it from there - like a chat on the phone. Most people are very happy to tell you all about themselves and love talking about the subject they rang you about.
If the person who rang enquiring about kittens is not comfortable on the phone, you may like to guide it with one or two open questions that requires more than "Yes" or "No" answers. Leave plenty of spaces/silences to give the person a chance to breath, gather thoughts, relax etc.
Yes, you want to be sure that the babies find the right homes but, no it should not feel like an inquisition or interrogation session.
Sometimes I just let them call again if they wish.
