Linguists will tell you they don’t study the correct way to speak a language, but rather the way it is (empirically) spoken. The rest of us, however, do abide by the rules (grammar, semantics) and view them as a social norm to be observed.
It is hard to pinpoint the exact reason why speaking a language ‘correctly’ is necessary or desirable; though one could single out social constructs as a reason to disfavor those who fail to ‘speak correctly’, hard objective truths as to maintaining correct form are hard to come by or prove. After all, we all know any modern language is basically a corruption / variation of older languages. Last decade’s mistakes are today’s rules, so is there a point to making a fuss about people’s mistakes?
Two easy examples are הַנְגָּשָׁה and literally.
The first, הַנְגָּשָׁה, has always annoyed me personally. (If you don’t speak Hebrew, I would skip this paragraph). Hebrew verbs are constructed via root and conjugation. The root נ.ג.ש. in the binyan of הפעלה already exists, namely הַגָּשָׁה. The nun was ‘swallowed’ over time into the gimel (explaining this is in English is kind of ridiculous); this is the meaning of גזרת חסרי פ”נ. In the fact, the meaning of הַגָּשָׁה is literally (snicker, see next paragraph) הַנְגָּשָׁה – to make something more accessible. For some reason in this present, post-modernistic world, people feel they have to make up a new word to convey a pseudo-new meaning. Well, the meaning isn’t new. “Making accessible” does have a modern liberal meaning relating to helping disabled people – be they blind of sight, have a language barrier, or in need of special physical access – but the core meaning (as in, “make this accessible to someone”) has always existed, even in the simplistic version of could you pass the salt? == make it accessible to me?, which is of course a common use of the word הַגָּשָׁה. So we don’t need this new, contrived creation of הַנְגָּשָׁה: we already a/the word for that (and it’s basically the same word).
The second example, one which might be more familiar to the English-speaking world at large, is the use of literally – or rather it’s misuse, as in “the movie was so good, I was literally glued to my seat.” To many people’s chagrin, this word seems to be gradually losing its meaning. I was enlightened to recently read on Quora a humbling perspective on this, in the same vein of the above – many words gradually lose their meaning, so no need to get worked up about just another word. The prime examples would be really, which literally means/reads “in reality” (and thus should be used the same way ‘literally’ is, but in effect is of course often used just to emphasize) and very (from the Latin root for ‘truth’; should be used for in truth but is of course just another tool for emphasis). So it seems literally may just be going down the same path – or more importantly, we use so many words in a non-literal sense that it doesn’t make much sense to get worked up about one more.
Despite any such objective legitimacy, however, it is hard to respect someone with a faulty grasp of their language. It seems the standard is social, rather than absolute; ‘correct’ language would such as spoken and understood by those around you – including any arbitrary rules observed or neglected.
Perhaps as in when in Rome, then – when speaking a language, do as the linguists do.