You can write this sentence two ways. Here we have to use 좋다 because 저 has the topic particle and 학생 has the subject particle. "As for me, the busy student is good" which in English we just translate to "like" anyway.
If you wanted to use 좋아하다 you have to change the particles. 제가 바쁜 학생을 정말 좋아합니다. When you like something, that thing receives the object particle.
The end of this sentence uses the word 좋다 which requires the noun (here, it's 학생) to have the subject particle. In contrast, 좋아하다 (which can be conjugated as 좋아합니다) requires the object particle.
In general, whether to use the subject or object particle (or other particles) for the corresponding verb/adjective have to be memorized. There are general rules to follow (like identifying if the verb is transitive or intransitive) but there are many exceptions.