If you wish to open Application Launcher upon clicking anywhere on the Desktop, for example, by pressing the Middle-Button of the mouse, you would normally right click on the Desktop, and select "Configure Desktop and Wallpaper...", go to "Mouse Actions", click "Add Action", press the Middle-Button of your mouse, and then select "Application Launcher" - that's all.
After this, when you press the middle-button of your mouse on the Desktop, you'll see the Application Launcher menu displayed, but a little differently then you would normally expect:  your applications are listed with their generic names (e.g. several apps listed simply as "Web Browser", which you can distinguish only with their different icons), and probably sorted according to the alphabetical order of their names, which might be another confusing factor. 
This can be fixed by going back to that configuration window, clicking on the settings button to the right of the Application Launcher, and selecting "Show application by name", which the only setting available, but unfortunately left deselected.
IMHO it would be much better if this setting could be selected/enabled by default.

Operating System: Debian Testing
KDE Plasma Version: 5.27.9
KDE Frameworks Version: 5.107.0
Qt Version: 5.15.10