Does grammar or punctuation really matter, though? How important is it to know whether a panda eats shoots and leaves, or whether it eats, shoots and leaves? I mean, how many of us have actually met a panda, let alone count one as a friend or acquaintance? Most people would assume a man eating tiger is a dangerous beast, not (as it technically means without the hyphen) an irresponsible human consuming an endangered species. And the only time it might be dangerous to leave out the comma from Let's eat, Grandma would be if Hannibal Lecter were in the room. (For some reason, all my examples involve food. I wonder what that means.)
I suppose it depends at what level we're trying to communicate. The average text message is probably comprehensible without any punctuation, but any attempt to communicate complex ideas of philosophy, artistic expression or law is going to need all the help it can get. The kind of misunderstanding that can be caused by a misplaced comma or a grammatical mix-up might not have such far-reaching consequences in novel as it could in an international peace-treaty, but it can annoy readers who aren't sure what you mean.
(And, for the record, my text messages are laboriously typed out in correct English, with proper spelling, full punctuation and grammar that's at least adequate for colloquial speech. Mostly.)
The important thing to remember is that grammar isn't just a set of arbitrary rules invented by sadistic scholars when they were bored: it's a fundamental aspect of language, and even of thought. There's a widespread view nowadays among linguists that humans are actually born understanding grammar, or at least understanding the need for it. Not the specific laws of a given language, of course, which may express grammar in any number of ways, but certain basic ways in which thoughts, and therefore eventually words, form patterns to create larger concepts.
Words are ideas. We may say, write or sign house, maison, casa or haus and mean essentially the same (not quite the same, since the idea will depend on cultural norms), but each word is merely a symbol of a specific idea we want to encapsulate. Language is the process of combining these symbols, and therefore these ideas, into coherent sequences of thought that can express anything from the location of a good food-source to the meaning of life, and grammar is the function each of these ideas plays in the construction of that concept.
We seem to fundamentally understand, for instance, the difference between identity and action (i.e. noun and verb), the relation in time and place of an event or condition to ourselves, and the concept of case. The cases of nouns and adjectives come in long and tortuous lists that must be memorised if you're learning languages like Latin or Greek, but case is essentially the understanding of what an idea is doing in the larger concept. Is it acting, or acted upon? Or does it stand in some relation to the ideas that are fulfilling those functions?
The expression of the functions, of course, can vary immensely from one language to another. The case function of a word, for instance, can be indicated by the word's form or its position in the sentence, among other ways. In English, dog bites man and man bites dog are fundamentally different sentences, because it's the word-order that determines which bites and which is bitten.
In Latin, on the other hand, word-order is irrelevant, because case is expressed by the form. Dog bites man could be equally canis mordet virum or virum mordet canis. To say man bites dog, you change the words, not the order: vir mordet canem or canem mordet vir.
Nevertheless, both languages are expressing exactly the same basic concept: that, to be at all useful, it must be possible to identify the precise function of each word.
The deep grammar we're born with is a matter of potential, not specifics, an instinctive need to look for the patterns in ideas which the brain then slots into place as grammatical rules. The rules will vary immensely depending on whether you're Japanese, Inuit or English, but some patterns seems to come up over and over, such as the tendency of grammatical functions to come in threes.
Three is a significant number in maths and science (after all, the world we consciously inhabit has three dimensions) and this seems to have spilled over into our linguistic understanding. There are three essential time-references or tenses (past, present and future), three genders in most languages that use them (although neuter has fallen out of use in the Romance languages) and three persons (I, you, he/she). There were even once three numbers, singular, dual and plural, although few languages now retain the dual. (In Classical Greek, it was used to express pairs — the eyes, the ears, salt and pepper, Laurel and Hardy.) Part of our shared deep grammar seems to be to find tripartite divisions.
Grammar is important, but specific grammatical rules aren't absolute. I'm currently reading a novel by John Crowley, a man who, I suspect, could probably recite the rules of grammar in his sleep, yet there are many sentences in the book that are punctuated "wrongly". The point is that each case is there to create a precisely calculated effect, such as leaving out the commas in lists to give the impression of ideas tumbling uncontrollably. Any rule of grammar can be broken — but only if you know precisely what rule you're breaking and why you're breaking it.
Then again, grammar isn't an exact science in practice. A good example of that is the problem of double negatives. The original understanding, in Shakespeare's time and still in languages such as French, was that negatives augment each other, whereas the consensus now is that they cancel one another out. In a way, it doesn't really matter which argument is more logical. Personally, I think the original concept makes far more sense, since the mathematical equivalent of the grammatical negative is zero, not minus, but language is about what its speakers understand, and it's generally understood today, for good or ill, that two negatives make a positive.
Being a writer is about trying to communicate a more precise meaning than is usually necessary in everyday speech, and grammar is a framework that allows all those shades of meaning to be brought into sharp relief. Getting a few grammatical rules wrong isn't the end of the world (unless you're writing an international treaty, in which case it could be) but it can make your work harder to read by leaving your meaning less intuitively clear to the reader. It can also lead to pandas being arrested on firearms charges, or fictional psychopaths eating your grandma. But perhaps that's a shade less likely.