To install:

Download and extract your bundle. This will be your AMPL folder. Rename it if you like and move it to any convenient location on your computer.

Next, you will need to tell macOS that your AMPL software is safe to run. Double-click your AMPL folder to open it and then follow these steps:

Control-click the ampl.command file and select 'Open' in the context menu.
If a warning message ("macOS cannot verify the developer...") pops up, click 'Open'.
A Terminal window will appear. After a few seconds, you will see ampl: in the last line in the window. You can now close the window. (If a warning message saying "Do you want to terminate running processes in this window?" pops up, click 'Terminate'.)
To run via IDE: In your AMPL folder, enter the 'amplide' folder and then double-click the Amplide application (with a black cat’s-head icon) to start the IDE. (The first time you open the AMPL IDE, you might see a few "Amplide would like to access files" boxes at startup; click OK in each box to dismiss it.)

To run via the command line: In your AMPL folder, double-click ampl. You should see an ampl: prompt appear and can now proceed to enter AMPL commands.