I believe the exclamation mark is not used in any different way in Korean than in other languages like English.
It expresses an emphasis, a surprise or other deeply felt emotions, which could include anger but it would be rare.
Also note that a lot of punctuation marks used on the Internet is nonstandard and not used in a serious writing.
For example, "~" is rarely used and repetitions like "!!" is not used in professional writing.
If someones says "그래! 난 이제 공부할 거야.", it is most likely expressing a resolve.
The person is saying "Yes! I will study from now on.", and "!" probably shows a strong determination in his/her heart.