Being sincere is being genuine. If someone compliments you on your outfit, but you notice them rolling their eyes when they think you’re not looking, they probably weren't being sincere.
If you've been offered a sincere thanks for something you've done, the person is probably really grateful. Sometimes when people say "thank you," they don't always mean it, but when they add the adjective sincere, the emphasis is usually for a very good reason. You may hear a charity sending out a sincere thanks to sponsors for their donations, which help keep the charity running.