You need to pass your Management Console password as an authentication token to every Management Console API endpoint except /setup/api/start.
Use the api_key parameter to send this token with each request. For example:
curl -L 'https://hostname:admin_port/setup/api?api_key=your-amazing-password'
You can also use standard HTTP authentication to send this token. For example:
curl -L 'https://api_key:your-amazing-password@hostname:admin_port/setup/api'