It doesn't mean, or translate to, anything. It's grammatical job is just to indicate that the following word is a direct object. To be grammatically correct, it's needed in this sentence. It is used only when the direct object is "definite": "the fish" as opposed to "a fish". You could say: "אני אוכל דג" "I am eating a fish" or "I eat fish". Then the "את" is not needed.
Omitting the "את" is actually a really common mistake that new immigrants make, you hear it all the time in Israel, especially around English speaking immigrants.
it's definitely a mistake but if you speak like that people will still understand you, even if the sentence isn't correct.
Yes, we can and do say it alone in sentences. It might not come up too often, but we do say it on its own. Suppose you walk into a restaurant with friends. One friend starts the conversation by saying: "I don't eat the beef here. It's too tough. I eat the fish, though." You respond, "I eat the beef but I don't eat the fish." And then the rest of the dialogue I wrote above. They're just trying to teach us grammar with short sentences, at first. Though maybe not used as often as longer sentences, they are still proper sentences.