it was actually very spontaneous but a bit complicated. basically, DSU nominated me as a student from their partner university (my home uni). the deadline was the beginning of june.
i had to send scanned versions of application for exchange students, medical check-up, personal statement, study plan, passport copy, transcripts from my uni validating my enrollment and displaying all my marks from all semesters. i also sent them a certificate of Korean language courses that i've attended. after they announced that i was selected, i had to prepare some more documents and make visa.
the only piece of advice i can really give to you is to write the personal statement as long as you can, write down the reasons you need to go to korea, what's your previous relationship with the country and so on. in the study plan, you just need to ground all your selected subjects.
i cannot tell if the scholarship is enough to cover my expenses, since my exchange will only start in the end of the month. but basically the only thing that you need to pay completely by yourself is the dormitory fee.