A nest is a place of warmth, comfort, and security, and a place that can be left and at times needs to be left.
Traveling with a couple of exiles on what could only logically be called a fool's quest, though thankfully Ik-Su's not a liar, and the world is more than simple, human reason, probably is more of a risk than need be.
Yet it's time for him to leave, he has to, and he has Ik-Su's blessing for this journey regardless, something to not be taken terribly lightly.
And so, Yun has to leave the nest, the place of warmth, comfort, and security, to go into the world without knowing if he'd even have any of those things. He won't have a stable place though, would have to give some things up, and live with greater difficulty, not that life was perfectly easy before. It didn't need to be easy to be home anyway, and home is beautiful, warm, and joyous, no matter what difficulty they'd have to face.
And yet life with a couple exiles and eventually Four Dragon Warriors can become a home too, with time, and even warm despite the cool of night.
It just takes time.