Before the NES, games were relatively short and arcade-like. The objective wasn't usually to reach an end-goal, but to rack up points until death. Saving games wasn't a concept until games started to change their goals. Games like Legend of Zelda, Metroid and Kid Icarus suddenly made sense to have save slots. But before that, they didn't really exist.
Then Nintendo added the battery back-up. Simply, they just soldered it to the inside of the cartridge, no memory cards or extra accessories needed. Unfortunately, these battery back-ups had a lifespan of about 15-20 years and are constantly fizzling out today, but the actual piece itself is relatively inexpensive and replaceable.
But in their prime, these little devices allowed Nintendo to be more ambitious with their games beyond gaining a high score in Donkey Kong. They could now embark on a long quest to save princesses in Hyrule or destroy Mother Brain.