Five times three is equal to three times five due to the commutative property of multiplication. Is there any interesting way to explain why this is so?
$5+5+5=15=5\times3$ and $3+3+3+3+3=15=3\times5$
Basically, if $15$ units were the length of a rod/sausage/stick then if you divide it into $5$ pieces of equal length (each of those pieces will be $3$ units long) via $4$ equi-spaced cuts OR into $3$ pieces of equal length (each piece being $5$ units long) via $2$ equi-spaced cuts, it doesn't matter. They'll all add up to give back the original length of $15$ units (obviously).
Bear with me for a second:
Multiplication commutes in $\mathbb Z$ because it commutes inside $\mathbb R$, and $\mathbb R$ has commutative multiplication because $\mathbb R\times\mathbb R$ is a Pappian plane.
In fact, the commutativity of multiplication in a division ring $D$ is characterized by Pappus's theorem holding in $D\times D$.
To me, this is an interesting reason for the commutativity of $\mathbb Z$ because it's rooted in the actual geometry of the real plane. Sorry if it is too obscure.
There are elementary pictorial or computational demonstrations (indeed, that is how I think the other answers are turning out so far) but I feel like they are not satisfying enough to be interesting, however useful they may be.