What's a `<script type='application/ld+json'>{jsonObj}</script>` in a `head` section do?

In your example, the script element is used as data block, which contains JSON-LD (type="application/ld+json").

JSON-LD is a RDF serialization. It allows you to publish Linked Data (or structured data) using JSON. In your example, the vocabulary Schema.org is used ("@context": "http://schema.org").

This structured data can be used by any interested consumer. Prominent consumers are the search engines Bing, Google, Yahoo, and Yandex, which support structured data that uses the vocabulary Schema.org. One use case they have is displaying more details in their result snippets.

Your example probably doesn’t lead to such an enhanced snippet. You have to check the search engine’s documentation if you want to know what features they offer and which structured data you have to provide for these. For example, Google uses the WebSite type (that’s used in your example) for their Sitelinks Search Box, but you would have to add a potentialAction in addition (for the search function).

It gives Google and other crawlers structured data about a website. This is used for rich snippets and knowledge graph panels among others. Have a look at this site for more information: https://developers.google.com/search/docs/guides/intro-structured-data

That's one way to include structured data in your site which helps any kind of users/crawlers use the information on the site in an efficient way. The most popular example is Google news cards:

This kind of card data are actually coming from structured data.

Other ways to include structured data is through Microdata

And the time of asking this question, I have no idea about these. Now I worked on structured data for some publishers.