Originally, this phrase meant a real physical battle where one could die, not just a tough struggle. Ancient gladiators would often fight to the death in arenas.
Pro Tip 2/3
Metaphoric Usage
The phrase can be metaphorical, meaning a fierce rivalry, not actual fighting. The debate team entered the tournament ready to fight to the finish.
Pro Tip 3/3
Not Always Serious
The phrase is often used in a non-serious manner to describe any competition, like sports. The two soccer teams are going to fight to the finish for the championship.