Krashen's third hypothesis helps us understand how Acquisition and Learning fit together in communication. It says "that when we produce utterances in a second language, the utterance is 'initiated' by the acquired system, and our conscious learning only comes into play later." 
How does this break down? When I want to communicate in Greek, it is the language that I have acquired that creates the speech. This is more obviously so in the case of spoken language production than written, but it is still the case in the latter. The 'Monitor' is my brain working to bring to bear language learning, that is my formal, conscious, explicit knowledge of the language's grammar.
So when I think ἔστι παῖς Ἑλληνικός ἐν τῇ στοᾷ it is what I have acquired that produces the utterance. The Monitor comes into play only when three conditions are met:
1. I must have sufficient time to use the Monitor. In free-flowing conversation this is rarely possible. If I attempt to use language learning in conversation I will have to mentally compose all my thoughts before speaking them, and the conversation will become slow and stilted. Alternatively, I may speak utterances, and then correct myself as I apply the Monitor post-utterance.
2. I must apply consciousness to form to use the Monitor. That is, rather than thinking about 'what' I am communicating, I must turn my attention to 'how' I am communicating. When I am doing composition exercises this is often the case - I will write down the Greek, and then rearrange or alter forms to be more grammatically correct or stylistically pleasing.
3. I must know the appropriate rule. That is, unless I know the rule, I can't correct myself. For example, unless I know, explicitly, that third person plural neuters in Greek regularly take a singular verb, I can't apply that to my utterances. If I have acquired that structure in Greek, I may intuitively do it, or I may feel intuitively that it feels right that way, but if I don't know the rule excplitly then my Monitor cannot apply it.
Point 3 helps to explain one phenomenon typically viewed as First language interference. When we don't know how to do something in a language, we will often attempt to supply the structure from our 1st language. This may succeed communicatively, especially if the rule is the same in the two languages, or if the speaker of the other language is familiar with our 1st language and so understands the structure we are trying to apply. Personally I have had this kind of interference across Indo-European and particularly Romance languages. When trying to think Latin if I don't know something then often the Spanish will intrude to supply the lack.
Clearly the Monitor interferes with language as communication. Yet it is not without its purpose. It is best and properly applied when composing written communication or prepared speech. Those are occasions when we do want to pay attention to form, have the time to do so, and either know, or can check, the rules.
How does this apply to classical languages? Most study programs are designed to build up language learning, and thus to equip the Monitor to do its job. Rarely, however, does such study lead to better acquisition, only to the illusion of acquisition.
 Krashen, S & Terrell, T. The natural approach : language acquisition in the classroom Oxford [Oxfordshire] ; New York : Pergamon Press ; San Francisco : Alemany Press, 1983. p30.