You can automate the web interface in Google the Google Maps "My Maps", but there is an easier way to get to the raw KML that you might not have known about.
For the current owner of the map within the Google My Maps, use a web browser to choose "Export to KML" and choose "Entire Map" in the dropdown, and choose the checkbox for "Keep data up to date with network link KML (only usable online)". That export will give you a KMZ file. Use an unzip program (7zip, unzip, Winzip, anything that understands zip format) to see the contents. Within the zip file will be a doc.kml file. Examine the contents of the doc.kml in a text editor. There is a <link> element which has within it a <href> element. That is the URL you can use with wget/curl/LWP to to your backup file retrieval.