We recently started using a third party software to do our in-house SQL backups so that the backup files are stored in a redundant and safe place. To confirm that the software was indeed compressing backups as it stated it would, we wanted to see what each backup size actually was in SQL so that we could compare that to what the software was telling us.
SQL stores lots of handy backup information in msdb in the backupset and backupmediafamily tables.
There’s some useful information in these tables, though make sure you clean them up regularly or else msdb can become enormous.