Why Won't a Database Backup if the Log File is Full?
BACKUP DATABASE is terminating abnormally
  
  Question: Why does the Database backup process require space in the
  log file?


Every data backup (full/differential) as the first step does a checkpoint. This is done for be able to grab as many as possible fresh data from disk.

Every checkpoint operation writes to the log about itself (at least 2 log records)

If your log is full even impossibility of writing checkpoint operation into log can fail your backup.

At the end of backup every full backup resets differential base, and this is also logged. 

And there are other log writes in between of this two, you can see them using sys.fn_dblog after doing successful backup. 

Here you can see the explanation of checkpoint and differential base:

Database Checkpoints (SQL Server)

How do checkpoints work and what gets logged

Base of a Differential Backup