How can I "clean out" my Time Machine backups if Time Machine won't backup because the disk is full? When Time Machine tries to make a backup, it informs me that it can't because the backup disk is full. 

What it's mostly full of (about 700GB of 1TB) is prior backups of the same machine. Time Machine used to just delete the oldest ones, but now it seems unwilling or unable to do so.

Is there some manual way I can make room by clearing out old backups?   When I go into the Time Capsule through Finder, the backups all appear to be in a sparse bundle, which I'm nervous to mess with.
Enter TimeMachine, then in the main window, select the backup you want to delete (on the right), then use the Action Menu (Gear icon) to select "Delete Backup".

Using this menu, you can also, when a file or a folder is selected, delete all its backup.