The Ramban (Bereshit 1:29) says the reason Noach was allowed to eat animals is because he maintained them in the Ark. If he would not have brought them into the Ark they would all have perished, so they owed their life to him. If so, why is he allowed to eat fish and not be like Adam who the Gemara says was not allowed to eat it? They were not in the Ark!
Iyun Haparsha had a list of at least 20 answers to this question. I can recall a few. Rav Chaim Kanievsky (some others as well) answer that once the issur of eating aninals went off so did the issur on fish because it was one issur. 2. Adam was never forbidden to eat fish. 3. Noach saved the fish as well. Since by the Teiva the water was cool. 4. Avnei Nezer says that really fish didn't become permissible to eat after the mabul. They only became permitted at Har Sinai when jews got the laws of kosher fish. Some add that fish became permitted then because if the jews wouldn't accept the Torah the world would've got destroyed, therefore they saved the fish. The Avnei Nezer adds that's why there's a custom to eat fish on Shabbos because that's when the Torah was given. This answer is also brought in the Pardes Yosef.
Noach's upkeep of the animals merited that he could eat from them afterwards. They, in a sense, owed him a debt (Radak and Ramban as quote by Torah Ladaat on Behaalotcha 'Achilat Basar').
Ramban (Bereishit 7:23) explains more about the fate of the fish during the Mabul. He brings a gemara (Sanhedrin 108b) stating that the waters of the Mabul were boiling and should have killed the fish as well. He offers 2 answers to why they weren't killed, both along the lines of the fish fleeing to cooler waters in the depths and elsewhere. This perhaps strengthens the question above.
I heard an answer (source?) that the Ark provided cool waters so that the fish could survive. This would explain why Noach would be allowed to eat fish as well; they were directly saved by him and his ark. The fish were in a sense 'saved by the Ark' as much as the animals were, thus permitting him to eat from them as well.