Because to be the head of a noble family they have to fulfill the condition of having an heir, and demons can only have heirs (babies) with swans, since they can't impregnate, or get pregnant by, the humans from their own world. Moreover, there is a plus in that the baby born from the union between a demon and swan will always be a demon like their demon parent). Having a child with a swan is the only way for demons to get the heirs they need, since all the other humans from that world can only give birth to demons if they are a pregnant woman accidentally influenced by the dark magic that happens only during the night of the red moon, but that is very a rare and random event.

Was it explained why Swans are hunted? Can someone remind me lol