First of all make sure backup compression is enabled on the SQL Server. You can execute below query to check the backup compression.
select name,[description],value_in_use from sys.configurations where name like '%backup%'
Execute below script to check the compressed backup file size and backup compression ratio.
Declare @FromDate as datetime
-- Specify the from date value
set @FromDate = GETDATE() -1
SELECT
CONVERT(CHAR(100), SERVERPROPERTY('Servername')) AS SQLServerName,
msdb.dbo.backupset.database_name,
CASE msdb..backupset.type
WHEN 'D' THEN 'Database'
WHEN 'L' THEN 'Log'
WHEN 'I' THEN 'Differential'
END AS backup_type,
msdb.dbo.backupset.backup_start_date,
msdb.dbo.backupset.backup_finish_date,
msdb.dbo.backupset.expiration_date,
DATEDIFF (SECOND, msdb.dbo.backupset.backup_start_date, msdb.dbo.backupset.backup_finish_date) 'Backup Elapsed Time (sec)',
msdb.dbo.backupset.compressed_backup_size AS 'Compressed Backup Size in KB',
(msdb.dbo.backupset.compressed_backup_size/1024/1024) AS 'Compress Backup Size in MB',
CONVERT (NUMERIC (20,3), (CONVERT (FLOAT, msdb.dbo.backupset.backup_size) /CONVERT (FLOAT, msdb.dbo.backupset.compressed_backup_size))) 'Compression Ratio',
CASE msdb..backupset.type
WHEN 'D' THEN 'Database'
WHEN 'L' THEN 'Log'
END AS backup_type,
msdb.dbo.backupset.backup_size,
msdb.dbo.backupmediafamily.logical_device_name,
msdb.dbo.backupmediafamily.physical_device_name,
msdb.dbo.backupset.name AS backupset_name,
msdb.dbo.backupset.description
FROM msdb.dbo.backupmediafamily
INNER JOIN msdb.dbo.backupset ON msdb.dbo.backupmediafamily.media_set_id = msdb.dbo.backupset.media_set_id
WHERE
CONVERT(datetime, msdb.dbo.backupset.backup_start_date, 102) >= @FromDate
AND msdb.dbo.backupset.backup_size > 0
ORDER BY
msdb.dbo.backupset.database_name,
msdb.dbo.backupset.backup_finish_date

