As far as my knowledge おかげさまで げんきです is a very polite reply to the question おげんきですか This reply means that, Thanks to God's or your good wishes and blessings, I am hale, hearty and fine.

I hope the asker and other viewers who learn Japanese to acquire the correct knowledge of the Japanese language. As a native speaker of Japanese, I am afraid to say this but I disagree with the opinion that "Okagesamade genki desu" is a VERY POLITE reply, because we use "Okagesamade" almost everytime we are asked "how are you?" In general, "okagesamade" implies gratitude for invisible assistance of many people around a person that brings about a good outcome, and we do not usually refer to God here. We use "okagesamade" not only in the reply to "how are you?" but also in many other expressions like the one posted by toru.

Actually, "okagesamade" does not have particular meaning here, and it is similar to "thank you" in "I'm fine, thank you". You don't actually mean to thank me for your being fine when you say this, but it is polite and nice to add "thank you". The same applies to "okagesamade".

Like Takashi said, there's no particular meaning in this context, but literally, "おかげさまで" can be taken to mean "thanks to you..."

As howdelightfull says okagesamade means literally "thanks to you". It is actually used to express one's gratitude when someone is helped, like "okagesamade mise no basho ga wakarimashita", meaning "thanks to your help I finally found the store." But in this context, unless the listener is a doctor or a nurse, there is usually no connection between that the speaker is fine and that the listener did something. So here it means nothing and the speaker just shows politeness. If I am spoken to this way by a non-native speaker of Japanese, I would feel he (or she) is not a beginner at least .