Danny, that's on a Google Nexus 4 which is Android. You just copy the file over to the phone, and then tap on the file (use a file manager app). If Google Earth is installed on the phone, it will open and zoom to the map.
It's a bit different with iOS, as there is no accessible file system. The easiest way in this case is to email the kmz file to yourself, open the mail on the phone and select "view attachment". This will open Google Earth to the map, assuming again that GE is installed. Note you can also use this method for Android.