"날 좋아하는 사람이 많아 / 날 좋아하는 사람이 많이 있어" => There are many people who like me. They are both correct sentences and mean the same thing. "많이(in great number) 있다(there are/exist)" is the same as 많다 (there are many). 많이 can of course be used with other verbs too: 많이 먹다 (eat a lot), 많이 주다 (give a great quantity). If the sentence ends with ~많아, it is usually translated as "There are many ... / Many ...". - 요즘 휴가 가는 사람들이 많아 [많이 있어]: There are many who go on vacation lately. - 시간은 많아 [많이 있어]: There is plenty of time. - 지진으로 죽은 사람이 많다: Many people died from earthquakes. 많아 and 있어 are just one of the four common ending forms: 많습니다/많다/많아요/많아, 있습니다/있다/있어요/있어.

Well, my impression is that the former is related to a speaker's judgement based on subjective/psychological criteria, whereas the latter the objective measurement. We have to take the intonation of each sentence into consideration though. I am afraid it is not helpful.