More than a Song
I think in the chanting of the Holy Names, we all have to go on a journey. It usually starts by us liking the singing of the Holy names. Who wouldn't like singing? Where I come from we would always sit together in the evening and sing. That is something the pop culture has stolen away from us and has given it to the stars. But we humans are singing beings, we like to come together in a group and sing. So it is maybe on this level where the appreciation of saṅkīrtana starts.
But there's much more to come. As you follow Caitanya Mahāprabhu’s teachings, you will learn more about the person whose name you chant.
He would say that chanting itself must be accompanied by hearing about Kṛṣṇa from the Bhāgavatam and from the other scriptures. At one point what you have heard and the Holy Name – the two begin to come closer. That means you chant Hare Kṛṣṇa and all of a sudden, by means of association, all of Kṛṣṇa's form, characteristics, His pastimes, they all become revealed.
This is why Śrīla Bhaktisiddhānta Sarasvatī Ṭhākura said the Holy Name should never come alone. Just like Kṛṣṇa is always together with Rādhā, the Holy Name must be always accompanied with His svarūpa śakti, His own potency, that is hearing about Kṛṣṇa from the Bhāgavatam.
Otherwise, it will only be a song.