Hey there,
One of our fellow Wix Studio users, Rita Maksymenko created an award winning anime themed website (https://design1go.wixstudio.io/anime) which features a grainy effect that is quite similar to the example website provided.
Upon further inspection using dev tools, I found out that it is a grainy image overlayed on top of the section, under the text and images.
You can use this technique as an inspiration and try building something similar.