To err is human!
One thing that I've learned throughout my career is that staying truthful and admitting your mistakes earns you the respect and trust you might not get otherwise.
I have been into situations where I made major mistakes and I could've easily put the blame on someone or something else, but I chose to admit it in front of everyone, and once that time was over, I felt the trust and respect coming my way.
Sometimes, it can be very tough to take in the criticism, you might feel broken inside, but trust me, it's worth it. Take it positively, correct your mistakes, and move on.
On the contrary, if you are on the other side of the table, try to give plenty of room to others to explain and justify themselves. Help them cope with the situation instead of making it worse. Don't forget, today it's them, tomorrow it could be you. No one is perfect!
Learn to admit your mistakes and empathize with others for theirs. At the end of the day, we're all humans!