Comment by potamic

It means they agree they are culpable in the specified activity, and are ready to accept its consequences. The consequences depend on the nature of the activity and the actors involved. If the activity was illegal, they agree to be charged with the crime. If the activity was unethical, they agree to be censured by their peers. If the activity was uncompassionate or immoral, they agree that their standing in community may be affected accordingly.

But most importantly, by assigning blame, they have cleared the contention at hand, taking the critical first step towards resolution of the matter. Without this, the issue may simply be stuck in limbo, taking up valuable time and resources and may never be resolved even. By acknowledging their mistake, they also wish to signal to others that they have learnt a peronsal lesson and would try to avoid making the same mistake in future. So it carries a lot of implications and is not an empty phrase at all.

Sure there may be people who throw around the phrase lightly, whether one believes them or not depends on how they are viewed in the eyes of others. So unless there is reason to believe otherwise, I would give them the benefit of doubt. If you want to imagine a world where the concept of "taking responsibility" doesn't exist, look at politicians. Is that how we want our society to function?