Why imports increase if higher relative inflation?
At first, U.S. exports increase because US exports become cheaper due to the low inflation rate. Inflation itself only influences the exchange rate indirectly through its effect on supply and demand for currency. That effect goes through trade. For that to happen, trade must first take place and so US exports must first increase. 

So at first the exchange rate doesn't adjust at all (no difference in supply/demand of the currencies just yet). US goods become cheaper so they're exported more. As US exports increase the demand for dollars by South Korea increases. This will later on lead to the south korean currency to depreciate and for US exporte to go down again (if the Marshall Lerner condition is fulfilled).