It's hard to say what the word 'literally' means. When I was in graduate school someone suggested that the literal meaning of a word is its most common use, and Wikipedia echoes this in part. But I suspect the most common use of the word 'literal' itself is incorrect and therefore not literal. If someone says "It was literally raining cats and dogs" they (probably) mean it was raining as much as it does when people typically say "It is raining cats and dogs." They don't mean that it was literally raining cats and dogs, i.e. that animals are falling from the sky. Literal meaning is better understood, I think, as non-figurative rather than normal or common meaning.
Why should anybody care? I suppose because of the people who say we should read the Bible and the US Constitution (etc.) literally. It's a political issue, in other words, since literalism is generally used to support conservative views. Against those who emphasize the (supposedly) loving spirit of the Bible or the liberal (and, e.g., supposedly pro-choice) spirit of the Constitution, literalists will insist on following the letter of the law. Along with rights to guns and no rights for gays, this leads to such additional benefits as the denial of evolution. So it's a big deal.
Having said that, debates about constitutional interpretation tend not to focus on literalism but on such theories or positions as originalism. For a response to that position I defer to Sean Wilson. One problem with taking the Constitution literally is that doing so doesn't give people what they want. The House of Representatives is not literally a house (not only is it not a place inhabited by a family, it is not a building at all but a group of people), the right to bear arms is not literally just a right to carry weapons but to use them, and talk of rights is hard to take literally at all--what is a literal right? Words like 'rights' don't have a clear, non-figurative meaning. When people say they believe in taking the Constitution literally they mean that they support something like originalism, which is clearly a kind of interpretation, one way among a plurality of ways of taking the Constitution. In this case, then, literalism does not get us away from interpretation. There is no option but to interpret (or ignore) the Constitution in one way or another.
Something similar goes for the Bible too. No sane person would deny that there is figurative language in the Bible. People who say they want to take the text literally don't really mean that. They mean they want to avoid over-interpretation (who doesn't?) and that they believe other people are guilty of such over-interpretation. This position faces problems, such as how to deal with the apparent contradictions in the Bible and what it could mean for the Earth and sun (and everything else) to have been created in six literal days, but these are not necessarily insurmountable. The point I want to make is that Biblical "literalists" don't actually take the Bible literally (i.e. completely non-figuratively). They support a particular kind of interpretation.
Does it follow that no utterances have a literal meaning, that all call for interpretation? No. However unlikely it might be, I could mean "It's raining cats and dogs" literally. Do utterances always need interpretation? No. They always need to be understood (if they are to be understood) but nothing is gained by calling understanding interpretation. Does understanding always take place within a context? Of course. Does this context matter to the meanings of words? Of course. Is understanding therefore always in fact a case of interpretation? No.
What about literature? Does a novel or poem have a literal meaning? I don't see why we shouldn't say so. Some literature is more figurative, some less so. "I caught this morning morning's minion" should not be taken literally: he wasn't out catching birds. But is he talking about a literal falcon or is this a metaphor? Or both? These seem like reasonable questions. It is useful to be able to distinguish a literal from a figurative reading, so why not allow for such distinctions? Is the literal reading not an interpretation? It is a way of taking the poem. And in the case of this particular poem, which is full of figurative language, it seems that any way of taking it is going to be an interpretation. But I don't see that anything is gained by saying that all takings are interpretations. If you recite a poem about an old man from Nantucket who has an unfortunate episode with a bucket, and I laugh then I am not interpreting your poem as funny. I do, I suppose, take it to be funny, but even that is a bit of an odd way to put the point. I just react.
Now, it might be objected that I don't just react. I react within a certain context, which has historical, political, social, etc. aspects. That's true. But it's also true that I react within a certain physical context: I am breathing air near the surface of the planet, etc. It's still true that I just react because "just react" means react-without-having-to-do-any-such-thing-as-stop-and-think. It does not mean that I react and do nothing else whatsoever, including existing within various kinds of contexts. Nor does it deny that these contexts affect the way (or fact that) I react. We shouldn't take the word 'just' here literally.