Its very natural that it confuse you! Of course you can say "私は肉を食べません"as"I don't eat meat." When you say"私は肉は食べません",It might be similar to say"I eat everything but meat."In this case,はof肉はis used to focus on what you really wanna say. For example :﹙私は﹚ネコは好きです(I don't like any animals but cats. ) ﹙彼は﹚今日は来ません(He doesn't come here today but might come tomorrow.) I'm not sure if you get it with my poor English. I hope I can help you!

は is originally a marker for emphasis(取り立て助詞）which has turned to mean "theme/topic" ここに花があります　vs　ここには花があります。 The meaning is the same; the ここ（に） in the latter is emphasized. は disappears when connecting with が and を (sometimes に）. 「このサイトがはitalkiです」→「このサイトはitalkiです」 「公園には花がいっぱいです」→sometimes→　「公園は花がいっぱいです」 So 肉をは食べません　turns to 肉は食べません 肉を食べません vs 肉は食べません equals to 肉を食べません vs *肉をは食べません The 肉（を） in the latter is emphasized, which implies "I rather eat fish, fruits, or vegetables" ※ は is a 係助詞 and を is a 格助詞; They are different in category of particle.

You can use either "を" or "は" in this sentence depending on the context. You usually use the former as you know. On the contrary, you use the latter when you specifically talk about that object. ／e.g. When someone asks you if you eat meat saying "肉を食べますか", you usually answer "肉は食べません" rather than "肉を食べません". In this case, you sometimes insinuate that you may eat other kinds of food such as "魚", "野菜", and so on.