You are running out of space probably because your server is in Full or Bulk-Logged recovery mode, and you are not doing transaction log backups.
First see what your recovery model. You can look it up in properties in the object explorer for the database, or run this query
If it is simple, you are running out of disk space because you are running some long transactions, and the space is it is using is actually required. If it is full or bulk logged, check if you are doing transaction log backups. Use this query:
SELECT database_name ,
FROM msdb.dbo.backupset s
INNER JOIN msdb.dbo.backupmediafamily m ON s.media_set_id = m.media_set_id
WHERE database_name = 'YourDatabaseName'
AND s.[type] = 'L'
ORDER BY backup_finish_date DESC
If you are not doing log backups (no records returned) or have not done log backups regularly, do a log backup. That should clear space in the log file, and stop the growth. Then, schedule regular log backups.
Look up some articles on log file management - for example here