Sincerity is a mix of seriousness and honesty. If you do things with sincerity, people will trust you.
Along with patience and compassion, sincerity is one of those qualities we all probably wish we had more of — and wish other people had more of, too. People who show sincerity are being serious, kind, and truthful. Politicians who show sincerity tend to do well because people believe them. An apology full of sincerity will probably be accepted. On the other hand, if you show no sincerity — by being sarcastic or dismissive — well, who wants an apology like that?