You don't need test management. You develop web applications. Your job is not rocket science. It's demanding, and you are doing a fantastic part shipping features and making customers happy. Who needs test management to do that, right?
Let's try three different points of view on this:
The software developer's Pov
You code and you write tests, even perhaps before writing application code? Different project managers come to you with their projects and ask you to be part of the team. You have a reputation with them that you are able to ship. You have your hands in several projects and can choose which ones you would like to work on. Your manager sets up an appointment for the yearly review. You want to ask for a raise. On what ground should you get your raise? Because other employees value you and your work? Because of your reputation?
What about this? You can click a button and produce a long report sheet that lists your contributions to several high-value projects. The report links your contributions with the JIRA ticket number, the commits that went into the feature and a detailed report of the tests associated with the feature/ticket. Your manager can also easily see, that tests for your code are successful in > 90 % of all cases, throughout the whole year. Compared to other developers you spent 19% less time per ticket and your bug tracker shows that code you wrote has the lowest propensity for defects.
The project manager's Pov
You handle two to three projects in parallel, know the backlog to these projects by heart even though each one is pretty long and try to manage your clients on which features will be released when. You are aware of contractual requirements and the need to comply with certain regulations. Your clients could ask you to produce a report on the software's quality and you would send it back within the hour. The client could see the user stories she created together with your team and she would also see that all of these were tested throughout the whole development cycle. She is satisfied and feels safe with your company since this report also made clear that you are doing regression tests for the integration with the legacy book-keeping system from IBM that sits in the client's basement.
The manager's Pov
Starting the project was a risky business decision. You are dealing with a heavily regulated environment. Any mistake could cost your reputation, but also your job and lots of money. You ensured the government official that your company would be up for the job because you have automated processes that document changes and outcomes. Risks are monitored and documented as well. The continuously running test suite is tied back into everything your employees create or modify, which gives you peace of mind because you don't fear the audit.
What's test management
A proper test management software will hold the project's requirements in an easily searchable way. You specify and create test plans that are linked to requirements. This link offers traceability. All types of tests are supported (Black-Box, White-Box, change-related tests, functional and non-functional tests).
There are JIRA plugins that'll offer that, but there is off-the-shelf software as well.
If you integrate this software in your workflows, you should make sure to write the proper tests, which will be how we continue next time.