"but how do they make the backups?"
Probably once using mysqldump, and once using the complete system-backup which accedently also includes the datafiles.
"...and must say, that the file copy is much safer (or i am missing s.th.) than the mysqldump. "
Indeed you are missing something.
Backing up the files seems safe, but requires that you lock the tables and preferably shut down the database before you start.
More to the point, backing up files means you backup everything, including any inconsistencies in the data. Should one of these inconsistencies prove to crash the database later, then restoring the data-files brings the datanase server back into the state it was in right before it crashed, which means it will soon crash again. Using mysqldump means you backup the SQL required to build the datafiles, and SQL cannot build inconsistencies.
One thing to note about mysqldump is that you should always do it from a crontab process, and never ever through a web-interface. And why because when you do it through a web-interface the process stops when you press 'stop' in your browser or when the page times-out (think of the infamous 30 second limit in PHP).