What does "No Thanks!" mean?

I always understood this to mean "no, but thanks for the offer."

Saying no would be a little blunt, so its just a polite way of refusing.

It is mere politeness. When somebody offers you something that you do not need or want, you just say "No, thanks!", thus showing that you are grateful for the offer.

In the brief context, it is almost certainly saying "No, thank you." in an informal way, since this is a discourse between two people and especially when one is offering something 'nice' to the other.

In general discussion, people will tend to be more explicit if the meaning is intended such as..

Yes I did such and such, no thanks to you!

As has been stated already, it is basically just a polite way to decline that became shortened over the years from "No, thank you" to "no thanks."

(Assuming Bob says "No, Thanks!")

Bob doesn't want any cookies, but he's showing that he's grateful for the offer. As opposed to just "No", it's the politer way to say the same thing.