Thanks for the post! This is expected behavior with the TestRail/Jira integration. When using the “add test case” button from within Jira, this will take you to a blank TestRail test case, with the only information being pre-filled being the Jira ID in the references field. In many cases, there are not direct analogs to the necessary fields within TestRail when coming from Jira, and often the test case details will differ from the requirement or issue from what is described in the Jira issue. As such, you would need to fill in the details of the test case manually.
The push template, which is what determines information that is passed to the Jira description, is used in the push function of the defects field when executing tests. In the page at Administration > Integration, you can pre-fill a push template with different variables pertaining to field data in your test cases and results, as well as static text or markdown formatting for use in Jira, and when you push a new issue during test execution, the data will be filled into the Jira description when the issue is created.
I hope that clarifies things!