As in English, you can't use countable nouns in the singular without an article - "She eats with girl" would be ungrammatical. For this reason, it has to be plural.
How am I suppose to distinguish between Madchen in plural form or singular form in this context?
Nope, that is ungrammatical. To be singular, the german sentense should read "She isst mit dem/einem Mädchen" to indicate there is only one. Just like in English, plurals can be used without an article: "I like fruits".
Yes, but we were supposed to write the word based on the audio. Don't "ist" and "isst" sound identical?
i was suppose to translate it. I think that it all depends on context. I'm sure that there is something in there that indicates what is going on, but i am too inexperienced with german to know what yet.