It’s mainly useful for a kind of continuous search, where the user flow is to search and get results, but not to necessarily navigate away from the results immediately.
I mentioned Twitter as an example. On their website you can search tweets by keyword, and if you leave the results page open (on the “latest” tab) then an indicator is shown whenever any newly posted tweets match the current search.
This could also be used to create media walls, that filter on some criteria and automatically update the display whenever new content is available.
These sorts of things could be achieved today by repeating a search on a frequent interval, but a push mechanism would provide faster delivery and use fewer resources.
For full disclosure, I’m not an Algolia user and I work for a company that provides push developer tools. So this is just a suggestion in case you find it interesting.