What is the significance behind the apples and boxes?

Question

In the show, the box and apple motifs reoccur throughout Mawaru Penguindrum.

What are the significance the two? Does it tie into the plot in any way?

Answer 1

I haven't seen it myself, but apples can often symbolize sexuality, temptation ("forbidden fruit") , fertility ("bearing fruit" as in your screenshot).

Connections can also be made to the stories of Adam & Eve and Sleeping Beauty.

The circular shape of an apple can also be a depiction of loops, or eternity - which is what it appears to be in Penguindrum:

It is revealed in the finale that the apple is a metaphor for the Penguindrum, which seems to be Kanba's life, shared between the three siblings

(TVTropes)

This article is more indepth about the apples in Penguindrum