Why would a dragon leave the clan to go adventuring?

[dragsaw]

So with the dragonborn in 5e the clan is more important than anything, life it's self, the gods etc. So why would a Dragonborn willing leave their clan to become a adventure?

[JoeJ]

They aren't necessarily leaving the clan, except in the purely geographical sense, as somebody leaves their spouse and children to go to work. The clan is still every bit as much a part of who they are as it ever was.

[Cespenar]

To return with riches, knowledge, power, etc. in order to protect it better?

To that effect, it would be good RP if a dragonborn would return occasionally and "put some money into the account", so to speak. Depending on the character, of course.

[StoicLeaf]

Ooooh I got one!
I use this for one of the NPCs in my game.

He and dozens of other Dragonborn are selected for a mission of utmost importance.
They are to wander Faerun in search of new lands to settle; the conflict with the Untherites isn't something either side can keep going forever and if peace cannot be achieved then only eradication or moving remain as options.
To aid them in their search they are all given a magic book which has a duplicate that remains in Tymanther: writing in one produces the same result in the other.

And so the scouts wander and write down their findings as they come across them.
Although not explicitly stated by the clan elders, the scouts are expected to wander until they find suitable lands.
Serving the clan in this way is honorable. Although one has to wonder how such an individual will fare, so far away from home, particularly when faced with the prospect of never seeing home again and dying in the middle of nowhere.

[TheCleverGuy]

Tragic backstory: The clan was scattered or wiped out by an enemy and your PC believes he/she is the sole survivor.