I have come across several times reading Chinese that they repeat words twice and then again.
For an Instance what is the difference between 热闹 and 热热闹闹 ？
what is the grammar meaning when words get multiplied in that manner？
As a native Chinese speaker, I've never learned the AABB "grammar rule" from book. This means it's never considered as a kind of grammar rule at least in my student age.
But I don't agree with @gcd0318. There is difference between AB and AABB. In Chinese, repeating usually means emphasize the AB word, or to make the sentence sounds good to phonology.
e.g. 勤恳（diligent and conscientious） vs. 勤勤恳恳 Although they have the same mean, but use 勤勤恳恳 always make the audience feel that the speaker is highly applaud one's diligent and conscientious. Use 勤恳 doesn't have the strong emotional feeling.
Back to your example, 热热闹闹, it's the same as usage of 勤勤恳恳. It emphasize how bustling it is.
And see this great answer for usage of repeating words in Chinese.
I do believe there is a differential meaning in multiplied words. One important meaning is that to repeat a word or phrase in the AABB manner is to impose a fictive duration upon AB. For example, we can speak 热闹一下，but not *热热闹闹一下 because 热热闹闹 denotes a duration reading that is incompatible with the perfect meaning expressed by '一下'.