I don't understand, because if 'don't' is 'do not' then wouldn't the sentence without conjunctions be 'do not you dare' which I'm pretty sure isn't grammatically correct. Like, when we say the term it means something else, certainly not 'do not you dare' anyway. Any ideas on why? I am so glad English is my first language.
TL;DR (Too long, didn't read): The “you” in “don't you dare” is emphatic repetition, but the first “you” is omitted due to the imperative mood implying the second person (you).
You is commonly used as an intensifier, often with the imperative mood. A good example of this is “you [noun] you”, in which the sentence would be fine without the “you” at the end, but the “you” gives the sentence emphatic repetition. In your example, “don't you dare”, the verb “do” (as in “don't” - “do not”) is in the imperative mood, so saying, “Don't you dare!” means that the desired action is that “you don't dare”. Notice how I added a “you” at the beginning, and removed the “you” after the “don't”. The “you” at the beginning is the subject, which is implied in the imperative mood, and the other “you” is simply emphatic repetition. In other words, the “you” in “don't you dare” is emphatic repetition, but the first “you” does not show up in the imperative mood. This is also explained by @1006a in a comment:
Parents (and their children) may be familiar with you-as-intensifier, at least in the US: "Don't put that in your mouth" becomes "Don't you put that in your mouth!" by the fifth repetition, and it might be "Don't roll your peas around on your plate" but it's almost always "Don't you roll your eyes at me!" It's not just for negative imperatives, either; it's also seen in sentences like "You stop that this instant," "You leave your sister alone," and (with a trailing intensifier) "You ingrate, you!"
The other sign that this is happening is that the word “you” can be safely removed. For example, the sentence,
Don't you dare touch that stove!
has the same meaning as the sentence,
Don't dare touch that stove!
These two forms can be used interchangeably, but the former is slightly more emphatic, and the latter is slightly more formal, so the form used can depend on the context.