why can we interchange summations

why can we interchange summations

Suppose we have the following
$$ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$$
where all the $a_{ij}$ are non-negative.
We know that we can interchange the order of summations here. My
interpretation of why this is true is that both this iterated sums are
rearrangements of the same series and hence converge to the same value (as
convergence and absolute convergence are same here).
Is this interpretation correct. Or can some one offer some more insightful
interpretation of this result?
Please note that I am not asking for a proof but interpretations, although
I would appreciate an insightful proof.