Translation:I am not busy this month.
Adjectives that end in い (often called i-adjectives) are made negative by dropping the い and adding くない.
いそがしい --> いそがしくない.
Other adjectives (often called na-adjectives) just add じゃない to become negative.
好き --> 好きじゃない.
There are a couple exceptions. いい is the word for "good," but it evolved out of the word よい, and that form is still used for negative/past conjugation.
いい --> よくない.
That's by far the most common exception, I believe.
It has to do with the fact that いそがしい by itself already means "it is busy": all i-adjectives in Japanese carry a tense/auxiliary verb "to be". That means there is no other form of "to be" in this sentence. And like other i-adjectives, the negation of this always takes the form of いそがしくない, not いそがしく いない （いない is the negation of the stand-alone verb "to be" for living things）. So, its polite/formal form is therefore いそがしくありません rather than いません, regardless of whether you're using it to describe a day/one's schedule or a person.
You're right. But in the case of ある，if 忙しく （isogashiku) is considered an adverb, then is 忙しかった （isogashikatta) also an adverb? And 忙しくなかった （isogashikunakatta)? The 'conjugated' form of the adjective are used in the same grammatical structure as the 'non-conjugated' adjective, so visualizing them as adverbs just doesn't work for my brain.
Nope, this form is correct too. Did you know that "aru" is an irregular verb? I didn't, until about half a year ago. Do you know what the plain version of the polite "arimasen" is? It's "nai". yes you read right, it's not "arunai", just "nai". This is where the provided "isogashiku-arimasen" comes from; it's really a polite conjugation of "isogashkunai".