Significance of modular arithmetic in DFT?
This answer to a differently worded question answers your question also.

And this answer is relevant.

And this answer and this answer.

I confess that I take great exception to anyone who denies that the Discrete Fourier Transform (and inverse DFT) fundamentally periodically extend the finite set of data presented to it.  I'm a nazi about it.  The DFT assumes that the $N$ samples of $x[n]$ passed to it are one period of a periodic sequence where $x[n+N]=x[n]$ for all integer $n$. And the inverse DFT does the same in that the $N$ samples of $X[k]$ passed to it are one period of a periodic sequence where $X[k+N]=X[k]$ for all integer $k$.

The modulo arithmetic done to the indices of $x[n]$ and $X[k]$ simply guarantee this periodicity while also guaranteeing that neither index, $n$ or $k$, go outside of the bounds of the finite sized arrays in memory representing $x[n]$ and $X[k]$.  That is

$$ 0 \le n = \operatorname{mod}_N(n+mN) &lt; N $$
and
$$ 0 \le k = \operatorname{mod}_N(k+mN) &lt; N $$

for any integer $m$.