I once came across a book called English Grammar for Students of French.
It seems like an interesting approach, based on the assumption that a better understanding of English grammar (your own) will give you a better understanding of French grammar too.
I think this makes sense. It seems a number of native English speakers find it difficult to understand grammar at all, mostly because it wasn't part of their school curriculum. Getting familiar with the notions related to the structure of sentences (the grammar) with words you know (in your native language, or a language you speak fluently) seems like a good starting point before moving on to a different language.
EDIT: (Adding a few points and reacting to other answers.)
My original point was more concerned about grammar than innately. (You can't really learn something innately, by definition of innate. Nevertheless, let's assume you want a more natural grasp on grammar.)
There are suggestions that you should read poetry and listen to music. I'd entirely disagree with that. Music and poetry tend to sacrifice grammatical correctness to make the words fit into the rhythm. What happens in English (mistakes like "I can't get no satisfaction", just to pick one) also happens in French. Even for texts that are on the more correct end of the spectrum, it's often quite difficult to determine where the sentences start and end across verses. You may also end up with learning constructs that are awkward or archaic, despite being correct. I'm not saying that you shouldn't listen to poetry and songs, but that they're not what you need for grammar; they're might do more harm than good in that respect.
Living in a French-speaking country is good, but that's mostly to get accustomed to the environment and encourage you to try to "think in French", if you can. On its own, it's not sufficient to learn grammar and to speak and write the language correctly. If you have long-term immigrants around you (in your native country), it's quite likely that most of them can speak your language but a number of them will still make a number of grammatical mistakes that you would easily identify. Granted, natives certainly make grammatical mistakes too, but it's a usually different set of mistakes.
Past a certain age, you just won't pick up grammar naturally only by ear, even if immersed in a country that speaks that language: you need to be proactive, have someone correct you if you can (or self-correct, which can be quite difficult, but sometimes doable). I'll re-iterate that learning your own grammar should help, if only to give you a certain degree of awareness of the structure of the sentences.
This being said, grammar on its own shouldn't be a purpose. Learning the language should be a balancing exercise between grammatical rules and actual content. Try to analyse the texts that you read in French using the grammatical rules you learn in parallel. The more you do it, the more constructs you'll pick up.
Be selective in the reading, audio and video material you use, at first at least. Everyone makes grammatical mistakes, but some make fewer than others. Reading newspapers like Le Monde seems reasonable, for example; reading comments on a random YouTube video in French is less. I don't watch it regularly, but France 24 (the French version) seems to have a reasonable combination of text and videos. (Reading the news of the day in two languages can help too, since knowing the bulk of the story should help you fill in the blanks.)
Regarding videos, one thing that I found particularly useful when learning English was to watch DVDs in English, with English sub-titles, so as not to "cheat": this way your brain doesn't fall back to your native language. Plenty of English-speaking DVDs (at least popular releases) have these sub-titles (sometimes with hearing impaired comments, but it's easy to ignore those). I found it harder to find DVDs of French films with French sub-titles unfortunately (even on the French market): French DVD publishers don't seem to care that much about the local hearing impaired (despite the fact that these subtitles do exist, since they're often available on satellite channels).