That's the thing about the LoZ games - is that the in-game lore, that does not get explained in the plot, can only really be theorised at best, as in some cases, there are either conflicting stories, or simply only enough to string together a theory of your own.
In this case, I would dare say that the conversion of the Goddesses to Dragons was simply to get them more involved in the game, without creating any confusing cross-overs.
So, what we do know is that the Goddesses Farore, Din and Nayru created the world - the Realm that the LoZ adventures reside in. And when they left the Realm, this created the Triforce. There is a fourth Goddess - Hylia. She was entrusted by the first three to protect the Triforce.
She then created several Dragons to protect certain regions of the world. Faron (not tied to Farore, or Farosh), Lanayru, Eldin and Levias. This then sets them in to the world and history of Hyrule - dragons exist because Hylia made them.
Dragons have existed in many games since... The Adventure of Link, OoT, OoS, WW, Minish Cap, TP, SS, and now BoTW and ToTK. But very little explanation on where they came from, aside from the above mentioned creation.
Farore, Din and Nayru are also mentioned rather frequently in game, but rarely more than in conjunction with the Triforce, aside from OoT and SS where they are mentioned in regards to the creation of the world, and OoA and OoS where they appear as the Oracle of Seasons, Ages, and Secrets. In the latter, they also are not Goddesses - merely people with the ability to use the item(s) associated with their powers - the Rod of Seasons, the Harp of Ages, and the Password System on the GBC.
Now, here's where the theories start.
Based on the importance of the Characters, Farore Din and Nayru likely would be involved in the games, but having more statues would probably only confuse things? So design-wise, adding them as dragons would likely be a good choice, but this then asks the question of their names - are they tied to the Goddesses? Or simply named after them?
Well, again we know that the time between the Original Timeline and BotW is a very, very long time. Ancient history. So much time has passed that civilisations have come and gone, been lost and literally buried under the stone, before the events of BotW begin and the remnants of these civilizations are found again.
So, based on this I have 2 theories:
- Since the Godesses exiting the Realm of Hyrule created something (i.e. the Triforce), it's a potential that re-entering the Realm at some point, for some reason had a similar impact. It could be that the Goddesses re-entered the realm, which turned them into Dragons (and/or created the relevant items). The Zonai then came to worship them as gods, and based on myth and legends, named them after the ancient Goddesses.
- Following the events of TotK, we now know that you can become a Dragon. It's likely the three Dragons were created in the same manner, again in relation to the Zonai - perhaps they came into being during the time of the Zonai, or the Zonai came across the item(s) and used them intentionally, perhaps believing that the power of the Goddesses were trapped within.