I have an old friend who has used that expression for years. I have long puzzled what it really means. I think I have it finally figured out and I thought you might like to know and share the secret, provided you keep it confidential!
You may not have noticed but you just finished reading a tiny example of the meaning.
I sincerely told you about sharing the secret with you, providing that you keep it confidential. While I was sincere about sharing, obviously I was not really expecting you to keep it confidential, but I was going through the motions of confidentiality for the record.
There are many other serious uses of this thought which are interesting and often essential and helpful.
In the "good old days" of fixed exchange rates when a country might be ready to change its rate of exchange, if speculators could get any advance knowledge, they could make bundles of money unfairly. Accordingly, when and if the Minister of Finance is challenged to reveal his plans, it was essential that he had to literally lie through his teeth until the actual moment of the change. He or she would say, "Absolutely no change is under consideration with respect to our rate of exchange!" He/she hoped their strength and sincerity was clear enough to avoid misuse of the information. When later accused of being a liar, the answer was simple: my duty required that I conceal the plan until the very moment that it became effective. For serious public servants whose influence is measured by credibility that is a tough problem to have.
Doctors frequently face a similar problem. If a patient is afflicted with a medical condition which is virtually certain to lead to death in a relatively short period of time despite all the doctor could conceivably do, what should the doctor say and do? Obviously the answer should vary to meet the circumstances. That said, many doctors, even in today's era of openness and frankness, try to be gentle and not deliver a final death sentence. Instead they talk about statistics and will to overcome odds to transfer the responsibility for reaching the ultimate conclusion to the patient, who will process all that bad stuff in their own way. The doctor can say sincerely that one should hope strongly that the odds will favor this patient, even if he believes otherwise. Fiddling with peoples' vastly different ideas about mortality requires great skill and deftness. Doctors do have to be sincere even when they are not certain!
A few years back an old friend had moved to a new community and was complaining about how unfriendly it was. He was on the verge of moving again. What should he do? The advice he got was, "It takes time to become part of a new community; give it a few more months." A couple of years later he was raving about his first move and asked--how did you know? Answer: I did not know, but from an objective point of view it made sense to give it time, even if that turned out to be wrong. Again a sincere, supportive answer was called for even if the outcome remained highly uncertain.
Two fishermen in their separate boats passed nearby. They reciprocally yelled, "How's your luck?" One yelled back, "Never worse!" The other answers, "Cheer up; it is great where I have been!" The secret is to manage expectations. Everyone knows that fishermen exaggerate and tell tales. Both these guys were sincere, and they both were hoping that it would get better at least for himself.
These examples are all either harmless uses of this type of sincerity in the sense of "little white lies" that smooth human relations. There are, sadly, other uses as well which are clearly antisocial. "Who me? I would not hurt a fly!" when accused of kicking a dog. Or the defendant in a murder case where he had killed his parents who said: "Take pity on me. I am an orphan!" Or the mother of an accused drug dealer says: "My baby boy has never done a wrong thing in his whole life!"
The trick in this kind of situation--whether it is a positive or negative use of sincerity-- is to examine the context of a statement from a source that may hold the cards you want to see and/or who has a stake in the outcome.
Then the key question is to ask oneself what you might have said if the situation were reversed. Putting your imagined thought against what your source says can give you a good start in triangulating what the source really believed as well as wanted to believe.
In all cases BEWARE!