They both arrived in the city 'within days of each other' = One arrived in the city only a few days before or after the other person. The elderly couple died 'within days of each other' = One died and then the other died just a few days later. Usually 'within days' is 2 to 7 days, but it is subjective.

It means that 2 events happened very closely together in time - for example, if a woman gave birth to a baby on August 5th and her friend gave birth to a baby on August 12th, someone could say that “they gave birth within days of each other” or “their children were born within days of each other”