> 跟着她“鸡”，“猫”，“狗”地念起来。

“鸡”，“猫”，“狗”地 here works as an adverbial phrase modifying the verb 念. 

“鸡”，“猫”，“狗” are basic noun words learnt by a beginner. Imagine the scene her 丈夫 following her reading these basic childish words, and you would get the idea of “鸡”，“猫”，“狗”地. 

Another point is that they don't put 鸡，猫，狗地， but “鸡”，“猫”，“狗”地, because the quotation marks indicate those words are said by someone. For example, 他说： “我今天会来。” The quotation mark is used to quote what one has said.