@Andrew's answer is correct, however just adding my perspective.
While 'really' is indeed an intensifier, I do believe it is often used as a way to soften the impact of a statement. Ironically, the effect is almost to 'de-intensify' a statement which might be considered to be giving bad news.
To add emphasis:
I really don't want to go to the party.
To remove emphasis:
I don't really want to go to the party.
I know I would say the second sentence if I wanted to avoid a very bald statement such as
I don't want to go to the party.
By adding "I don't really" the sense is softened, perhaps by adding a degree of uncertainty. Other adverbs act in a similar way - "particularly" and "especially" spring to mind. "I don't really want to go to the party [but I will if it makes you happy/but I could perhaps be persuaded to change my mind]."
In essence, what we are saying when we use these adverbs after one of the verbs is:
"I don't want to go to party with any great enthusiasm".
Logically, we are saying "We do want to go to the party, but only a tiny amount". Thus, it softens the let-down for the person who hears the sentence because we are kind of avoiding saying we don't want to go at all.