Why is $n$ divided by $n$ equal to $1$?

$\frac ab$ is, by definition, the solution of the equation $bx=a$. Thus $\frac nn$ is the solution to the equation $nx=n$. Assuming $n\neq0$, this equation has the unique solution $1$.

In higher math, you usually take the equation $n/n=1$ as the definition of division by $n$. Specifically, $1/n$ is defined to be the unique number such that $1/n$ times $n$ is 1. Then you go on from there to define $2/n$ and other numbers using ideas called equivalence classes.

So it's that way because mathematicians feel like it. Like Bill Thurston said, math isn't real, it's just a way of organizing human thought.

You don't need division. $\ b^{n+1} = b\, b^n\stackrel{\large n=0}\Rightarrow b = b\, b^0\,\Rightarrow\,b(b^0\!-1) = 0\,\Rightarrow\, b^0 = 1\,$ if $\,b\ne 0.$