How do I exclude existing Magento categories from Algolia?

Do you mean that you want to hide the categories from the autocomplete section?

Or, do you mean that you want to remove certain categories from being indexed? 

If you mean you want to hide categories from the autocomplete section, you can set the number to 0 to disable it.  (Stores > Algolia Search > Autocomplete Menu)
 

If you mean that you want to remove certain categories from being indexed, there are two ways to do this. You can either add a filter to all your front end queries that exclude the relevant categories, or you  can prevent them from being indexed altogether. 

You can add a filter to your search widget by following this guide for customizing your front end in Magento. You can do this with the searchFunction in your instantSearch widget, adding filters or facetFilters using NOT or "-" respectively. If you choose this option, ensure that you use the provided hooks and  Magento best practices.

If you choose to not index them, you can use the guide here and note what products are indexed, and toggle those settings to prevent them from being indexed.