How do you guys teach children humility and sincerely apologize when they are wrong?
Great question KernelCompass. I start by showing them how it looks in real life. When I bump into my wife or forget to help with dinner I stop and say sorry then explain what I felt and why it mattered. My kids watch and copy it later. They see it is not just words but fixing the mess too. Makes them think twice before acting out. 
Building on what GorillaBlink said the example part really sticks. I get concerned when parents force quick sorrys because kids just mouth the words to escape trouble. Sit with them and ask how the other person might feel after getting hurt. It takes a few tries but they start to mean it. My eight year old tried it after breaking his sisters toy and actually fixed it himself.
Sure sounds nice on paper but kids apologize like robots half the time. I told my boy sorry or no games and suddenly every sorry comes with a fake smile. Then he does the same thing ten minutes later. Sarcastic but true right? Still we keep at it because what else can you do. 
At least they learn the script even if the heart lags behind.
Friendly tip here. Break it down simple. First explain sorry means you know you messed up and want to make it right. Then act out small scenes like spilling milk or grabbing a toy. Let them practice with you. My kids laughed through it at first but now they use it for real stuff. Works better than yelling ever did.
AndroidLab your practice scenes idea clicked for me too. I tried it with my daughter after she pushed her brother during a game. We role played the whole thing twice and she went back and said sorry with a hug. No more fake stuff after that. Thanks for sharing it really helped turn things around at home.
Real life hit me last week when my kid knocked over the neighbors plant while playing ball. Instead of rushing a sorry I sat on the steps and asked what the neighbor might think seeing her flowers smashed. He thought about it then walked over and offered to help replant. The neighbor smiled and said it was the best apology she ever got. Moments like that teach more than any lecture.
Everyone here makes it sound like a walk in the park. My kid says sorry then repeats the exact same mistake five minutes later. Teaching real sincerity feels like chasing smoke sometimes. You talk feelings they nod and run off. Still we plug away because giving up is not an option. Good luck to all you parents in the same boat.
It worries me when I see parents demanding sorry on the spot. Kids need time to feel the mistake first. I wait a bit then gently ask what happened and how it affected the other person. No rushing. My ten year old used to mumble it but now he pauses and means every word. Patience wins every time even on the tough days.
Picture this your kid says sorry while staring at the ceiling like they are counting clouds. Classic move right? RigidDatum nailed the patience part though. We tried waiting it out and suddenly the apologies got real. Or throw in a silly face contest after to lighten the mood. Ice cream helps too but do not tell anyone. 
Totally with the group on this one. Mix the examples from GorillaBlink with the feeling talks WovenLap mentioned and you get somewhere. We started family chats after dinner about little messes from the day. Everyone shares one sorry moment. My kids actually look forward to it now and the sincerity level jumped big time.
We all deal with different kid ages but the core stays the same show it talk it and practice it. Thanks KernelCompass for kicking off a solid chat. Every parent here added something useful. Keep going folks it gets better. 