I guess op meant "vec(ABC)=kron(B.T,A)vec(C)", and my attempt at explaining it would be:
If you take the result of transforming the columns vectors in the C matrix by AB and vectorize it you get the same as vectorizing first C and then transforming it by a block matrix obtained as the Kronecker product of B transposed and A.
The significance is that it performs a reduction of matrix calculus to vector calculus (i.e., it shows that you can convert any matrix calculus operation/formula/statement into a vector calculus operation/formula/statement).
Comments
I guess op meant "vec(ABC)=kron(B.T,A)vec(C)", and my attempt at explaining it would be:
If you take the result of transforming the columns vectors in the C matrix by AB and vectorize it you get the same as vectorizing first C and then transforming it by a block matrix obtained as the Kronecker product of B transposed and A.
The significance is that it performs a reduction of matrix calculus to vector calculus (i.e., it shows that you can convert any matrix calculus operation/formula/statement into a vector calculus operation/formula/statement).