The following options are correct, and nearly synonymous:
You get an extra 15 days.
You get another 15 days.
Both "an" and "another" act as indefinite determiners.
"You get an extra 15 days" relies on the "extra" between "an" and "15" to be grammatically correct. Without the "extra", and with "days" as a noun, the two determiners ("a"/"an" and "15") conflict. But "You get a 15-day grace period." is grammatically correct, because "15-day" is acting like an adjective.
Even though "another" is short for "an other", "You get an other 15 days" is not grammatically correct.
There are contexts where "You get the other 15 days" both makes sense and is grammatically correct. These contexts require defining both a set of days, and another set of 15 days, before referring to "the other 15 days". For example, "You have 25 days to walk across Nevada. You can spend the first 10 days in the mountains, and you get the other 15 days to cross the desert."