There are a lot of polyglots posting videos on the Internet. This guy is pretty remarkable, for example. As is this one. Here is a multilingual conversation between two more, Luca Lampariello and Richard Simcott. And here is Mr. Lampariello on his own, talking about grammar. It felt really, really nice to see someone who knows a lot more languages than I do making the same point about grammar that I've made in this blog post 3 years ago. Don't waste any time studying grammar while learning languages! You'll either get it gradually and subconsciously, while trying to make sense of large volumes of spoken and/or written material, or you won't get it at all. From Luca's video:
"I don't think we need to be aware of how it works. We just have to make it work. Once you're able to speak a language, make it flow, and once a language is accurately expressed, you have internalized grammar unconsciously."... "It is not a subject to study, it is an ability to learn."
One of the reasons why the conscious approach to grammar doesn't work is that the grammatical rules one encounters in mass market textbooks are gross simplifications. The system is always many times more complex than they let on. What about specialist linguistic literature then? Some of it may succeed in covering most of the complexity of a natural language, but only by exacerbating a different problem. You don't want to have to think about convoluted sets of rules and long lists of exceptions to them in the middle of conversations. This stuff has to be available to you immediately on the subconscious level, like the fine muscle movements that you'd use to keep your balance on a bike.
Why, then, is grammar taught and written about? Mr. Lampariello mentions one of the reasons in his video: to a scientifically-inclined mind grammar can be an interesting subject. Not because it can help such a mind learn a new language, of course. Complicated systems can be interesting to nerds by themselves, even if there's no hope of doing anything useful with them. And one can actually achieve useful results through the serious study of grammar. For example, it can shed light on the relationships between different languages, and through that, on the history of human migrations.
The second reason why grammar is taught and studied is much more prosaic than the first: there's a sucker born every minute. The most effective way to learn a foreign language is to immerse oneself in it - speak with natives, read with a dictionary, watch a lot of TV. Not much breathing space for teachers or textbook sellers here. But we can't have that. There's always going to be someone offering to teach you language X for a fee. Grammatical rules can help such people claim that they're teaching a proper subject, something akin to chemistry or accounting. They're not.