"I am a little busy this month."
Adverbs are like adjectives for verbs. They change the meaning of the verb (eg. quickly, slowly): 忙しい busy is an adjective: I worked at my busy office. (<- This sentence implies the office was busy.) 忙しく Adverb of busy is busily: I busily worked at my office. (<- This sentence implies that "I" was busy.)
(I'm still an amateur at Japanese, so don't ask me to use those in a sentence, haha.)
But in the sentence it is not using the adverb form but the negative form of the adjective for conjugation
The negative form of the adjective is adverb + Nonexistence.
To make an adjective negative you must attach it to the negative verb of existence "arimasen" formal or "nai" informal.
You use the adverb form to describe the state of non-existence. "Is busy" in negative literally becomes "does not busily exist"
So 忙しい Busy → 忙しく Busily → 忙しくない Not busy(informal) / 忙しくありません Not busy (formal)