Apple - How do I make an AppleScript file into a Mac App?

In Script Editor, select File > Save As and set File Format to Application. (In Mojave and later, the Save As choice in the File menu is not visible unless you hold down the Option key.)

Open Bundle Contents drawer and see the applet.icns file, copy your icon to that.

See "Saving a script as an application" in Script Editor help.

In El Capitan at least you can use osacompile, e.g:

osacompile -o MyScript.app MyScript.scpt

to export an an application, do File -> Export and set the drop-down list to "Application"