piece by piece 作方式状语。根据韦氏高阶，此用法中 by 的意思是：used in phrases to describe how a series of people or things proceed or are dealt with（用于短语，表示一群人的行进方式或一连串事物的处置方式）。类似用法再如：

They entered the room two by two. 他们两个两个地走进房间。

I solved the problems one by one. 我把问题一个一个地解决了。

We checked the list carefully, item by item. 我们逐条仔细核对单子。

Police searched the area house by house. 警察挨家挨户地搜査该地区。