Exporting tests can be done in different ways: either by using some of Jira's built-in features (which vary from Jira version to Jira version) or by using Xporter app. What you can do with Jira out-of-the-box may be rather limited but it may be enough to address your needs. If you want to have full control over the layout and the format of the generated document, then Xporter is the way to go.

Using Xporter app to export to fully customizable PDF, Word, Excel, CSV documents

By using Xporter app, you can fully customize the layout of the document containing the specification of your test cases. It also gives the ability to either generate documments in .pdf, .docx  or even .xlsx and .csv formats.

You can use some of the already available and free Xporter templates as basis, by downloading them, customize to your needs and upload to your Jira instance. These templates can be found in Xporter's Template Store, under "Tests & QA" section. 

Some good examples for making a document related with test specification are the "Xray Test Set" and the "Xray Test Report" templates. There are some other ones, more related with execution, such as "Xray Offline Test Report" (if you're interested on this topic, please check out this blog post).

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