There are quite a few intended meanings of these qualifiers; here are some examples of possible meanings of "I'm sorry," with each interpretation in parentheses.
"I'm sorry, can you tell me where the library is?" (Excuse me...)
"I'm sorry, but that doesn't work for me." (I hate to have to tell you this...)
"I'm sorry, but no." (No matter what you say...)
"I'm sorry, what did you say?" (I beg your pardon...)
"I'm sorry, I just couldn't get there on time." (I have a good excuse if you want to know...)
"I'm sorry, I said vanilla, not chocolate." (Were you paying attention...?)
"I'm sorry, I totally screwed up." (Please forgive me...)
"I'm sorry, but she went to jail?" (I can't believe what you're saying...)
"I'm sorry, I'm busy right now." (Stop bothering me...)
"I'm sorry, I tried to stop him." (It's really not my fault...)
I could come up with even more, but I hope I've made the point that English is tremendously flexible, and that context, inflection, intonation, the situation, and the demeanor of the speaker, all influence meaning to the point that the same two words can mean dozens of different things.
Best Answer
It is an example of adding emphasis, and being "extra polite", as Josh61 says in his answer, but I think there's some extra subtleties which are worth pointing out.
When we interview someone in front of an audience, and reach the end, we (the interviewer) need to emphatically signal to our interviewee and the audience that the interview has ended: this is the interviewee's cue to relax and the audience's cue to applaud or whatever. Simply saying "Thank you" is not a strong enough cue for this purpose: it needs to be something that catches everyone's attention, and something that you would not normally say during the interview.
Combined with a particular rise-and-fall speaking pitch, ("med med hi-hi-hi low low") "Thank you very much indeed." works very effectively: the "indeed" acts like a very clear sign saying "The End".
In this sense, the phrase "Thank you very much indeed" is actually operating on several levels at once: it's an English sentence but it's also an aural cue operating on a level beyond language - almost like music. Even someone who didn't speak english could learn to recognise it as a "This is the end" cue, and it will tend to alert even the people in the audience who aren't paying good attention.
Even if there's no live audience, an interviewer may still adopt this technique, partly out of habit but also to tell the viewer at home and the interviewee that they've reached the end.