EDIT: Since the introduction of Combat Manuals, there is no longer a reason to send heroes home.
Before, there was no other reason to send heroes home besides for feathers or to keep the barracks from getting full.
Now, converting heroes into Combat Manuals works just as well to keep the barracks from getting full, and can be exchanged for feathers at exactly the same rate as sending heroes home.
Combat Manuals are a way to get rid of heroes (especially duplicates), while still keeping them around for skill inheritance, or until you decide to use them for merging. As such, sending heroes home is now nothing more than a shortcut for creating a Combat Manual, and then exchanging the Manual for feathers.
As for why you'd want to turn heroes into feathers:
It's one of the fastest ways to get feathers when there is no event currently going on, like Tempest Trials, Forging Bonds, and so on.
In the arena, you can earn up to 7100 feathers per week. You're much more likely to earn between 1000 and 2000 feathers, though.
As you keep summoning heroes, you'll eventually end up with lots of 3-4 star heroes, which you might not want to keep if you've already decided on which heroes you want in your team. By sending them all home, you can earn between 750 and 1500 feathers per 20 Orbs, assuming you summon the full set of 5 heroes per summoning session.
You'll need all these feathers in order to improve the rarity of your heroes. A 3 star hero needs 2000 feathers to become 4 star, and a 4 star hero needs 20 000 feathers to reach 5 stars. Some heroes can not be summoned at 5 stars and therefore need to be improved using feathers.
If you're not afraid to spend money for orbs, sending heroes provides a limitless source of feathers (limited only by the amount of money you're willing to spend).