Question

How can a change in the temperature of an ocean current affect the climate of a nearby coastal area?

Answer 1

change in temperature of an ocean current , has total relations with change in pressure formation in adjacent lands.
if the current is cold, the the pressure would be high and vice versa.

when a change takes place it does not only change the weather of the place but the weather system across the globe.

Take the case of El-Nino. It is warm periodic ocean current which replaces cold Peruvian current every 4-5 years. This results in floods and torrential rains in Peru. This also causes delay of monsoon in Indian subcontinent which halfway across the world.
It has drastic effects not just in the nearby coastal area but weather system of Earth.