They elaborate on that in the rest of the series. That is actually not filler, but more directly from the actual canon material itself. Since we're in Spoiler Talk, I guess I'll be more blunt on the subject.
"THE KING" is just the future generations of Konoha. It is a metaphor based off of Shogi, which is essentially a form of chess. In chess, the objective is to protect your own king while trying to corner your opponent's king. In the truest sense of the word, 'the king' basically means something that one must protect. For Asuma, his father, and the rest of the people that believe in the philosophical 'Will of Fire', the most important thing a person in Konoha could ever do is to always protect the future generations to come. Shikamaru realizes this and his new objective in life is to protect Asuma and Kurenai's child and become it's mentor, teacher, and friend much like Asuma was to him.