"Claiming responsibility" for something means saying that it's your fault. You're the one who did it.
I claim personal responsibility for the mistake, and I promise you that it will never happen again.
"Claiming responsibility" is usually a noble and mature thing to do. However, there's another sense of "claiming responsibility". When there's a terror attack, a bombing, etc., sometimes a group will "claim responsibility" for the attack. This means that they're saying (truly or falsely) that they did it.