It is required to make sure that all the necessary configurations were made.
First of all, please check what search engine is selected for your site in default Magento settings:
Stores > Configuration > Catalog > Catalog > Catalog Search > "Amasty Elasticsearch"
Secondly, check the general settings of the extension:
Stores > Configuration > Amasty Extensions > Elastic Search > Search Engine > "Amasty Elasticsearch"
Then, click "Test Connection" button and make sure that you see "Successful! Test again?" message.
If all of the above is set up correctly and search still gives no results, please create a support ticket with us as there might be a conflict with some third-party extension in your site.