First of all this is not a method, but an approach to teaching and learning a second language. The emphasis is on meaning versus the more traditional approach that emphasizes structural/grammatical competence. A communicative approach focuses on authentic and meaningful exchange of new information. Teachers pose genuine questions as opposed to display questions. An example of a display question is "Are you a student" versus a genuine question is "what does your uncle do for a living?" The emphasis is on authentic materials versus a traditional textbook. By using authentic materials, students see the connection to their own language and culture as they have the background knowledge to access this knowledge. Communicative approaches emphasize negotiation of meaning, social contexts of learning, interaction among students, information gap activities, cooperative learning, role playing. There is an attempt to replicate the immersion of a natural language learning environment. Grammar is an important of the classroom, but it does not drive the curriculum.
This subject is very interesting. Some teachers or investigators think that this would be an effective method of learning, whitout grammar, only giving the language as we do to our babies. It sounds natural and less difficult. Personally I think that any method will take you to the goal, the question is with how much effort and time. This method seems to be similar to learn to drive whitout learning the rules of traffic before you start the engine, the teacher that sat next to you will tell you every time you reach a give way signal, "stop!", ok you say, next time, the same thing, "stop!", ok you say, so at last you will find out that you have to stop at every give way signal, but you will not know why, that's the problem. My question is, wouldn't it be easy just tell the student: "Look, you have to stop every time you approach a give way signal"?
I don't claim to be an expert, but I have been doing quite a bit of reading and practicing the communicative approach. I think what you are really talking about when you say "...method of learning, without grammar, only giving the language as we do to our babies." is not the communicative approach, but the natural approach. While Krashen's natural approach did appeal greatly to common sense, and thus had an effect on the way we think of language instruction, the communicative approach does allow for grammar instruction. However, it is very clear that grammar should not be the main focus of learning. Instead, we should be teaching students how to communicate, and as the grammar becomes a related topic, we teach it. It is not that we only teach grammar by correction, or that we hand out notes on verb conjugations or pronoun usage rules. It is about finding real ways to communicate so that when they are learning rules and uses of language, it makes sense.