The word "sorry" does not always signify an apology. For example, when a person says "Sorry about the rain," they do not usually mean "I am sorry for having caused the rain." Instead, they mean "The rain caused me sorrow." It's of little surprise that recognizing a shared sorrow increases feelings of trust.

A: "Sorry about your leg." B: "It's okay, it's not your fault."

It's sympathy, not an admission of guilt.