In my point of view, grammar can be learned on later, but for learners of a language(other than children under the age of 8), grammar is essential to become fluent. You can start out learning simple phrases without learning grammar and start conversing in a language, but grammar is needed to create your own sentences in my opinion. So start out with a phrasebook and then once you get a feel for the language(intermediate level), start studying grammar.

The important thing is to eat baked beans every day. Nothing else matters. If you eat baked beans every day, your English will be perfect on about 3 months.

grammas is everywhere not only when you write, but also when you speak. Some scholars say that the name grammar need to be changed into understanding the language

Grammar is important. I have a guess. This is what I believe. Your teacher did NOT say "grammar is unimportant." You misunderstood. Your teacher means that it is good for YOU to work on fluency first, grammar later.

The simple fact is that if you don't exercise proper use of grammar, people will think you're stupid. Also, it could lead to confusion, but that isn't a huge issue, you could always explain yourself if need be.