Given a cubic cake, defined as $\{(x,y,z)|0\leq x,y,z\leq 1\}$.
We cut it by the planes
$p_1\leftrightarrow x=y$
$p_2\leftrightarrow y=z$
$p_3\leftrightarrow x=z$.
How many pieces will we have after cutting?
And the 4-dimensional case: $\{(x,y,z,u)|0\leq x,y,z,u\leq 1\}$
How many pieces will we have after having cut by the spaces
$x=y$, $x=z$, $x=u$, $y=z$, $y=u$, $z=u$?
And the $n$-dimensional case...
When cutting by the space $x=y$, we divide the cake in two parts:
One part with $x>y$ and one part with $x<y$.
The same holds for spaces like $y=z$, $z=u$, ...
That means, that every ordering of the variables $x$, $y$, $z$, ... defines a part. And since there are $n!$ orderings, there are $n!$ pieces after cutting.