고3 (read '고삼') means 고등학교 3학년 (the third and last grade/year in high school).
The official school and schools years in Korea are as follows:
- 초등학교 (primary school) => 6 years. 1학년 (초등학교 1학년 -> 초1), 2학년, ... 6학년 (초6).
- 중학교 (middle school) => 3 years. 1학년(중1) ... 3학년(중3).
- 고등학교 (high school) => 3 years. 1학년(고1) ... 3학년(고3).
The abbreviated words like 고3 can also mean the students in that year, i.e. 고3 학생 or 고3생 = high school senior.
So "고3 화이팅입니다" is like "Fighting! (High school) seniors!".
The senior year is the most stressful year because the students have to take 수능시험 (SAT) in addition to many other tests, which more or less decides which universities they can go to (and thus has a great impact on their future). So the phrase is probably meant to boost the students' morale.