I’m not as well-versed in Korean as I hope to be. But below is what I understand (please take with a pinch of salt):
개 gae = dog (having watched so many Kdramas, we know there are variations of dog-related swear words. Adding ‘dog’ to it makes it all the more ‘derogatory’, in korean terms.)
새끼 saekki = young kid, young dog(i.e. puppy)
자식 jasig = child/offspring
Hence, ■■■ gaesaekki and also 개자식 gaejasig = SOB/b*stard.
Anyone, please feel free to correct.