Why Korea becoming multiethnic?

Because many reason.

First, Many people come Korea to work.
Many citizen of Korea don't want to work hard thing called 3D (Dirty, Difficult, Danger)work.  Employer need to find worker but not easy in Korea.
So they hire the other country people. Like south eastern Asian.
They come Korea and get a job , will marry Korean. This is one of major reason

and Second, because of rural men.
In modernization, many people come city and live there because convenience and a culture life.
Like that process, also many women come city and hard to find women in rural and don't want to live in rural. Men in rural want to marry but can't find partner in their place.
So many people turn eye sight to the other country. This is easier than other.

Why the other country women come to Korea? Major reason is money.
Most of rural men is old. They are older than the best age. but who marry with them is really young than other marry women. Because easy to get money than their country. So they're sacrifice for their family and come Korea with old guys
Nobody can't say this is bad thing... Just this is what we faced.