It just means there are two possible cases depending on the person's birthday, not two different ages for the same person. Suppose you're 20 years old in Western age and your birthday is January 15th. Then you have already gained one year in 2018 in the Western age, and since everyone gains one year in the beginning of the year, the two ages are synchronized so that the difference is only one (20 vs 21). But if you're 20 years old and your birthday happens to be February 1st, you have not yet gained a year in 2018 in Western age like you have in Korean age. The ages are out of sync and the difference is now two (20 vs 22) until you pass your birthday and becomes 21. This means if you know someone's Western age but not their birthday, you can't tell their Korean age. It could be one or two more than their Western age.

An interesting age calculator: http://waegukin.com/korean-age-converter.html

Koreans get a year of age on the first day of every year not on their birthday. And when a baby is born they earn a year as soon as they are born. so if you are born on the 23rd of May, 2015, you are already one year old when you are born, and then you become 2 on the first day of 2016. Therefore, throughout the year 2016, your age is 2. Then on the first day of 2017, you become 3 years old until the first day of 2018. Basically Koreans become older all together on the first day of a new year.