What is the purpose of 「ん」in this sentence 「忙しいんだよ」?
There are two differences in pronunciation and time tense.
At the points of pronunciation, 「いそがしいんだよ(isogashimdayo)」's pronunciation is softer than 「いそがしい(isogashii)」, because many Japaneses image that streaks of consonants make sentence more informal and softer.
(Note: いそがしいんだ「よ」, last 「よ」makes so informal expression in Japanese.)
At the points of tense, 「いそがしいんだよ」 is seemed to be present perfect tense. In other words, 「いそがしいんだ」 translated in English, "I have been busy." There are some emphasis of CONTINUITY.
But 「いそがしい」is seemed to be present tense. In translated to English, "I'm busy." There are no emphasis of CONTINUITY.