Ok, there are many answers to this question and many have some truth to them. However, there is one major difference between saying I AM BORED and I MISS YOU VERY MUCH.
When Russians indicate how a person feels (e.g., hot, cool, interested, bored, happy), they use the dative case of the person plus the adverbial form (-о) of the complement. So I AM BORED in Russian would be МНЕ СКУЧНО.
Now your sentence: Я ОЧЕНЬ СКУЧАЮ has nominative form of the personal pronoun I and it really means: I MISS YOU VERY MUCH or I TRULY MISS YOU. "You" is implied and does not need to be present in the sentence, it is usually clear from the context.