You’re going to want/need to customize the instantsearch.php template file that we ship with the plugin.
https://github.com/WebDevStudios/wp-search-with-algolia/wiki/Customize-your-search-page
You’ll want to start with this area and create a div to store the taxonomy widget output:
https://github.com/WebDevStudios/wp-search-with-algolia/blob/main/templates/instantsearch.php#L32-L39
Then you’ll want to add the new widget similar to this section highlighted below, depending on what output type you want to match.
https://github.com/WebDevStudios/wp-search-with-algolia/blob/main/templates/instantsearch.php#L167-L182
For example, if you want them to match the behavior of tags, you’d want to duplicate and edit this one
instantsearch.widgets.refinementList({
container: '#facet-tags',
attribute: 'taxonomies.post_tag',
operator: 'and',
limit: 15,
sortBy: ['isRefined:desc', 'count:desc', 'name:asc'],
}),
You’d need to change the attribute line to match your taxonomy name.