Luminar is available for purchase from the Mac App Store, and the Skylum Store. The update flow is different for different versions.
If you have purchased Luminar from the Skylum Store, launch the software, load any image, and click on LUMINAR from the top menu. Then click CHECK FOR UPDATES.
If your version is up to date, you'll get the message that says "You're up to date". If the new version is available, you'll see a screen explaining what's new and the button to install the update. Click on the button and complete the update process. Updates are free for all the current users.
You can also set-up automatic updates with the non-App Store version of Luminar. When updating to the new version, choose Auto Updates when asked.
If you have purchased Luminar from the App Store, all the updates will happen through the App Store. To check if the update is available, launch the Mac App Store application, and click Updates from the top bar.
You'll see a list of pending application updates: find Luminar and click the Update button. Here's a very good Apple guide on how to update your Mac OS applications.
Download the full Luminar User Guide in PDF Format