When I say learning grammar, I mean it in an active sense. Obviously to become more fluent in a language, a language learner has to learn grammar, but a lot of time that is done passively. They're learning it because they have memorized the patterns they hear from their daily practice with speakers of the language they are trying to learn. In other words, they're learning the language communicatively and adoping the proper grammar without understanding the rules behind it. Exposure to people speaking the language and memorizing the common phrases they say is far more important than studying grammar, in the beginning. Language learners should aim to increase their interactions with native speakers rather than allot a significant amount of time to learning, say, tense patterns if they want to gain fluency in a new language.
This is just my opinion, though, based on personal experience. I speak many languages because of exposure. I've spent more time studying Japanese grammar than I have the grammar of any other language--in the beginning, that is--and I've struggled most with gaining fluency in the Japanese language. I can think of no other reason than lack of exposure.
And it's not that grammar isn't important. It's just that no one needs to actively study grammar to learn a language.