The short answer
A branded hashtag is a hashtag unique to a specific business, product, or campaign, rather than a generic, widely used tag. It exists so a business and its community can consistently use the same tag to gather related content in one searchable place.
Its real job is organization, not discovery
Hashtags now function mostly as categorization rather than a direct reach driver on most platforms, so a branded hashtag's value is organizational. It gives a business a single, searchable place to find and reshare content its own community has created, more than a way to reach strangers browsing that tag.
Simple and specific outperforms clever
A short, memorable, unambiguous branded hashtag gets used by a community more consistently than a clever or complicated one. The tag has to be plain enough that people remember and correctly type it without checking.