I came across a simplified description of Gödel's theorem and the discussion touches on a concept of honesty (truth?) and completeness. How does Gödel's theorem apply to everyday interactions?
It may never affect your everyday life, but it has weakened our trust in rigid logical methods, as a culture. If even mathematics cannot attain to this kind of complete coverage of a domain, there is a good reason to think we habitually overvalue the role of rules in science.
I think that the shift toward seeing more of the human side of scientific inquiry, and admitting that it is deeply affected by personal faith, was unchained by the brake this kind of result put on logical positivism.
It is in effect the first post-modern fact. Even if you don't go down the whole trail of postmodernism, it keeps the bug in your ear that says absolute modernism strives for more than can be realistically attained. Sociology, faith, human nature, etc. really do matter in the end, and will not just be steamrolled by the sheer power of any system.
Here's what Jordan Ellenberg, a professor of mathematics at the University of Wisconsin, has to say about this topic in his Does Gödel Matter? article:
What is it about Gödel's theorem that so captures the imagination? Probably that its oversimplified plain-English form—"There are true things which cannot be proved"—is naturally appealing to anyone with a remotely romantic sensibility. Call it "the curse of the slogan": Any scientific result that can be approximated by an aphorism is ripe for misappropriation. The precise mathematical formulation that is Gödel's theorem doesn't really say "there are true things which cannot be proved" any more than Einstein's theory means "everything is relative, dude, it just depends on your point of view." And it certainly doesn't say anything directly about the world outside mathematics, though the physicist Roger Penrose does use the incompleteness theorem in making his controversial case for the role of quantum mechanics in human consciousness.
So the short answer to your question seems to be that it doesn't, and that extreme care should be taken not to misuse or misrepresent the theorems.
Edit: given the high number of upvotes this answer has received, I should point out that I'm by no means an expert on the subject, and that an alternative, more in-depth explanation by someone who knows more would be highly appreciated.
An almost real-life example of the simplified explanation you've referred to could be procedures in a huge corporation, if they are complex enough. Imagine a procedure:
A procedure that doesn't comply with The Company's mission must not be followed
Now, imagine a coffee-drunk, inexperienced employee at 5AM accidentally modifies company's mission statement by adding this sentence:
The Company doesn't allow procedures with description starting with the capital 'A'
Should now all the procedures that don't comply with company's mission (for example being obsolete, after earlier modifications of policy's mission) be followed or not?
This is of course an instance of the liar paradox. While this doesn't express the whole of Gödel's theorem it is closely related.
The described situation is not strictly real-life as it probably haven't occurred in reality :) However, systems of procedures may be viewed as formal systems, and when they become complex they often have problems with consistency and completeness.