How can I look like this?

How can I Google my website and the category looks like a picture above?
I'm using wordpress and the SEO plug-in is using 'Rank Math SEO'.
I'm also confused whether those are menus or categories.

Add description and keyword in the meta tag. It shows up in the search result.

Can't read Korean.

You need to implement structured data using Schema.org markup via the Rank Math SEO plugin. This involves enabling Schema Markup in Rank Math settings and configuring the appropriate schema type for each pagRe.

May be you are using the nulled themes or plugins and that is hacked,
It happens even if you have cheap hosting without much security ,
Check it and change the hosting if it is hacked and resubmit to search console

These are called "Sitelinks" by Google. No matter what the CMS is, google can find out the structure and different sections of a website and shows this. Sitelinks are not always visible to all users. Also, you website needs to have some authority for this feature to show up. In WordPress you can set this up with individual pages or category links. Google usually picks up links from the menu present in the homepage of your website. https://developers.google.com/search/docs/appearance/sitelinks