> 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.