The word "cardboard" appears in the very first line of the poem:
The cardboard shows me how it was.
This can be interpreted in one of two ways, but each will lead to the same overall meaning.
First, the "cardboard" can be used to describe the thicker paper upon which old photos were printed. Unlike the glossy and flimsy photo papers often used today in quick and convenient printing, old photos were often printed on a thicker, more sturdy backing—much like a thin cardboard.
Another interpretation is that the photograph is framed by a simple piece of cardboard. Through this cardboard frame, the photograph emerges to show the speaker the significance of the joy of these little girls.
Cardboard helps to establish setting and tone. First, the setting is in the fairly distant past, in a time when a cardboard was commonly used to either back or frame photos.
Second, the cardboard helps to establish a simple tone. It conveys a simplicity in the usage and shines the spotlight on the image it captures, the one that reflects the joy of her mother who has been dead now for as long as this girl in the photo had lived when the photo was taken.