MacBook – Time Machine says “There isn’t enough space” and says I need 50 petabytes available to back up a one terabyte drive

The sparsebundle is how a backup looks when it's a network backup. This isn't supported by Apple when you connect to an Airport which is why there is a minus symbol.

You can connect the drive over USB and start backing up normally but you won't have one place where all the backups exist (and worst case - you'll have two copies of each file - one in the sparse bundle and the other in /Volumes/whatever/Backup.backupdb as a flat directory backup instead of a sparse bundle format).

In practice, this isn't really a bad option since you only will lose a file when the drive fills and you could archive the sparse bundle or mount it to see if you can just delete those old backups. If you need to recover files, you can mount the sparse bundle and do the recovery by hand (or hand copy that one snapshot in time into the new format and recover things).

Lastly, you could move that sparsebundle to a supported destination like OS X server, Time Capsule or a NAS drive that works with Time Machine and continue backing up to the sparse image. These all would let you continue backing up, but cost time and/or money. I wouldn't recommend hacking OS X or Airport to make it appear as a Time Capsule, since it doesn't check the sparse bundle which will become corrupt sooner without some process tending it.

Time Machine is using differential backups. That means your first backup is a complete backup which is same size with your drive. Starting from the second one backup will include changed files since last backup.

These Backups are accessible by finder. So you can easily find out which folders causes these big backups. After you find the folders you can manually exclude these folders from Time Machine Backup Options.

Some softwares may cause huge backups if they changes files last updated status like Dropbox Agent or similar. You may consider exclude folders of these softwares because they are already copied to the cloud.