Algolia's Search API lives on your application's own hosts, e.g. https://<APP_ID>-dsn.algolia.net. Auth goes in two headers.
Copy your Application ID and a key from API Keys. Send them as x-algolia-application-id and x-algolia-api-key. In the examples they're written {{ALGOLIA_APP_ID}} and {{ALGOLIA_API_KEY}}.
Full reference: https://www.algolia.com/doc/rest-api/search/
curl -X POST https://{{ALGOLIA_APP_ID}}-dsn.algolia.net/1/indexes/{{INDEX_NAME}}/query \
-H "x-algolia-application-id: {{ALGOLIA_APP_ID}}" \
-H "x-algolia-api-key: {{ALGOLIA_API_KEY}}" \
-H "Content-Type: application/json" \
-d '{"query": "taco", "hitsPerPage": 5}'curl https://{{ALGOLIA_APP_ID}}-dsn.algolia.net/1/indexes \
-H "x-algolia-application-id: {{ALGOLIA_APP_ID}}" \
-H "x-algolia-api-key: {{ALGOLIA_API_KEY}}"