Make box for storing eggs primarily (6)
I believe the answer is:
create
'make' is the definition.
(I know that make can be written as create)
'box for storing eggs primarily' is the wordplay.
'box' becomes 'crate' (crate is a kind of box).
'for storing' means one lot of letters goes inside another.
'eggs primarily' becomes 'e' (I can't explain this - if you can you should give a lot more credence to this answer).
'crate' going around 'e' is 'CREATE'.
(Other definitions for create that I've seen before include "found" , "Make or invent" , "Produce from nothing" , "institute" , "Devise" .)