Translation:You are happy.
很 fills the spot for am,are,is when the word following it is an adj.
The difference between 你很高兴 and 你高兴 is that while the first one means "you are happy" the second one means "you are happier than..."
Adjectives by themselves work as verbs, in Chinese
It's a little more complicated than that and I wish I could explain more, but my english isn't technical enough to do so (not my first language)
In Chinese you connect noun + noun with a word 是 (to be). I am Serj. 我是Serj.
But you can't use 是 to connect adjectives, you need to use 很. And it doesn't really mean "very", it's almost completely lost the meaning in this construction, and only connect words, for example:
I'm good - 我很好。 It doesn't really mean "I'm very good", and you can't say it without 很.
There are stronger words you can use if you really want to underline that "I'm very / really happy" 我非常 (fēi cháng) 高兴
I don't know if it's just me but '你高兴' sounds a bit incomplete as if the speaker still have something else to add. I mean it is definitely correct when you incorporate '你高兴' into a sentence but you'll seldom hear people say it alone.
Just like what @varigby said, 很 or 好 will make you sound more natural :)
很 is simply grammatically necessary when an adjective is used as a predicate. (At least in "subjective", less factual sentences, as opposed to colors, nationalities etc.) Its original meaning of "very" is practically lost in these cases. (If you want to say very, really, truly indeed, you need something like 非常.) In short, this kind of sentence is incorrect without 很.
很 normally means very but when you are forming a sentence in the form "X is Y" where X is a noun and Y is an adjective, you need to add 很 anyway. So in this sentence the effect of 很 is a bit blurred, think it rather like a filler word. To convey the meaning "very" there are other particles like 非常高兴, or 太高兴了.
In English sentences, if the predicative is an adj., there is a "be/am/is/are" before the predicative, which doesn't exist in Mandarin. so there must be an adv. to fit in the spot instead, just as you said. The adv. can be 太(too)，很(very)，蛮(quite)，非常，好，有点(a little/bit)，有些，etc.. BTW, if the predicative is a noun, there still will be a 是(be) in mandarin.
You are very happy also makes sense...because 很 definitely means very in Chinese. However, in this case, you don't need to overthink this 很. In other words, if you try to speak Chinese, you want to express you are happy, it doesn't really matter "very" or just "normally", you always say 你很高兴
很 means "very," but as people use it a lot, its effect has faded. People usually just add it to a sentence out of habit and don't really intend to use it as "very." Unlike what several other comments say, you don't need something to connect 你 and 高兴 in this case to mean "You are happy." 你高兴 is correct. It's just that Chinese is pretty much a language of habit. To use 是, 你是高兴的 would be clearer than 你是高兴, as 你是高兴 can also mean "You are happiness."
In modern Chinese, a lot of words are compound words made of more than one character. For example, 高兴 is a compound word.
Similar to English, in Chinese, you can use a series of syllables instead of just one syllable at a time to guess what was said. For example, if you heard "my [ant]'s house," you can be almost certain the [ant] is "aunt," and if you heard gao1 before xing4, you can be almost certain it's 高兴.
However, also similar to English, it can be a homonym that's use less frequently. For example, the [ant] can actually be "ant," and the "gao1 xing4" can actually be 高姓 (高 isn't a rare last name in China).