King of the Underworld and ruler of the dead, Hades isn't known as a big softie. So it's a bit surprising that Orpheus is able to talk him into letting Eurydice go. But eloquent Orpheus finds a way to pull on the one tender string inside Hades' dark heart: his love for Persephone.
Hades had been so smitten with Persephone that he kidnapped her and forced her to be his wife (not the healthiest way to start a relationship, but he was kind of a twisted guy). You can read Shmoop's take on that whole story if you want some more deets. In any case, by comparing his love for Eurydice to Hades' love for Persephone, Orpheus successfully bonds with the grim god.
Of course, Hades doesn't take kindly to people who don't follow instructions, and he certainly doesn't give favors twice. So when Orpheus looks back a Eurydice after being explicitly told not to, Hades snatches her back to the Underworld without a second thought. Yeah, that's more like the Hades we know.