What does "the dragon has three heads" actually mean?

Source : Google

Well, the most thrown-out answer to what people think it means is that the three-headed dragon refers to Aegon the Conqueror and his sister-wives. Prince Rhaegar, on the other hand, seems to have interpreted it as needing three Targaryen children in order to fulfill a prophecy. Others think it simply means needing three dragonriders.

However, I think everyone in truth is misinterpreting the actual meaning of House Targaryen’s sigil, and, in the Targaryen’s case, they have forgotten the meaning of their own words: Fire and Blood

Let's be clear: there is no such thing as a three-headed dragon, at least not physically, so it doesn't simply refer to Aegon and his two sister-wives. No, the three-headed dragon refers to the dragonbond, as do the words of House Targaryen, fire and blood.

I am a proponent of the Lemur theory.

The idea is that the dragonbond is similar to skin-changer magic in that there is a spirit within the dragon. A sacrifice can be made to transfer the spirit of a human into a dragon. Thereafter, that human's descendants can bond with that dragon and its descendants; this is the blood of the dragon, if you will; hence, the fire in our souls and blood in our veins hold the secret to our bond (that's also why the Valyrians practice incest, to keep that bond within their own family so that other families couldn't ride their dragons; it was a power move).

This three-headed dragon is the spirit, the dragon itself, and the descendant who rides the dragon.

This also explains why no one was able to tame the dragon, Cannibal. Even the Targaryen's being the Blood of the Dragon wasn't enough because Cannibal was not of the same brood as the Targaryen dragons; he came from another Valyrian family who perished within the doom.

The House of the Dragon seemingly confirmed this theory with Daemon’s song:

Fire breather
Winged leader
But two heads
To a third sing

Daemon is acknowledging the dragon first, the fire-breathing winged leader of the human dragon pairing, then it says, But two heads, meaning there's something more going on here, which is that somehow two heads are singing to a third head.

From my voice:
The fires have spoken
And the price has been paid
With blood magic

So here Daemon wants the fire to speak through his voice, presumably to awaken the ancestor spirit in the dragon. The price for this power has been paid with blood magic, which means magic centered around human sacrifice, so that would be the price paid by Daemon's ancestor, who was killed and was spirit stuffed into Balerion’s great-great-great grandmother, and that spirit essence was passed on to the subsequent generations of dragons.

With words of flame
With clear eyes
To bind the three
To you I sing

This line makes it even more clear that the words of flame that Daemon is singing, through which the fire speaks, are in fact a spell that is intended to bind the three heads of the dragon. Two heads singing to a third is apparently itself an act of fire magic and blood magic because Daemon and Vermithor are awakening the sleeping third head of this dragonbond. The clear eyes bit simply means seeing things clearly, seeing the truth and the magic of the dragonbond.

As one we gather
And with three heads
We shall fly as we were destined
Beautifully, freely

Daemon is singing a song that is actually a spell to begin to prepare Vermithor for bonding, just as the lyrics suggest, and that as one we gather with three heads refers to awakening the invisible third head with spell and song so that all three heads can work in unison. That is how we shall fly free as we were destined, and that is how you fly as a three-headed dragon, not three dragons and three riders, but one dragon with three heads joined as one.

So when Daenerys hears in the House of the Undying that "the dragon has three heads," little does she know that she, at this point as a future dragonlord, is the three-headed dragon, who spiritually joined together with the other two heads, namely her mount Drogon and the spirit essence of her Targaryen ancestor, both of whom were present in her journey at the House of the Undying, and this bond has been forged with fire and blood.