“fathers.” Not just the people’s immediate fathers. Here, in typical Semitic idiom, “fathers” means ancestors. The Jews had been sinning for centuries, and the Babylonian captivity was a consequence for that.
“iniquities.” The Hebrew text reads “iniquities,” but in this context, “iniquities” are being put by metonymy for “punishment.” The children were bearing the punishment that their ancestors should have received themselves because they were the ones who sinned. This often happens in historical and/or spiritual settings. An ancestor does something bad that later affects the children.