OK after a lot of messing around I figured it out. First select all layers you wish to remove the background from and hit the Options button:
In the popup that comes up, hit the "Set Transparent Color..." button:
In the popup that comes up, select the background colour you wish to hide (black in my case) and hit OK:
You'll be sent back to this ...
Based on the comments the features which have the line through them are multipart features. These are single features with multiple geometries.
If the different parts of the multipart features are touching perfectly then using the Vectory geometry > Fix geometries should merge them together into a singlepart feature.
If that doesn't work then go to ...