The discrete Gaussian distruibution have the most entropy at the same standard deviation compared to other discrete probability distributions. It's important in some LWE/SIS digital signature algorithms because it helps reduces the signature size (most notably in BLISS and FALCON), but otherwise not so essential in key exchange or encryption (and has been replaced with binomial distribution in NewHope).
The reason using Gaussian distribution reduces the signature size, is because it allows certain entries in the signature to be represented with shorter codewords using e.g. Huffman coding. But doing it right can be difficult, as slightest observable difference from true discrete Gaussian distribution can lead to leak of private key information.
Take a Fiat-Shamir signature scheme as an example, the signature $z$ is calculated as:
$$u=Ay$$
$$c = H(u,m)$$
$$z = y+sc$$
Where $A$ is a component of the public key, $m$ is the message, $s$ is a private key component, and $c$ is the output of hash function and $y$ is randomly generated according to discreate Gaussian distribution.
We want $z$ also be in discrete Gaussian distribution, becuase
since we've added $sc$ to bind the private key to the signature, and
$c$ is a public component,
we have to hide the private component with $y$. And here, in order to achieve the expected distribution, rejection sampling is used.
See also: How would low-precision Gaussian sampling impact the security of BLISS?
