Pets offer lessons in life, and death. They're often the first introduction a child gets to death-- that mysterious, eternally heartbreaking idea of whatever life is being stripped away. We will never accept it completely.
A pet is our friend, one so innocent and so loved. When you know that the moment is soon coming, your heart is crushed and your eyes are wet. There's nothing you can do, and you know it, and you hate yourself for it.
Eventually, if death is not in a hurry but taking its merciless time, you do start considering quality of life. You think more about their suffering, not how sad you are. You force your weary feet to make that harrowing trip to the vet, and return with a shoebox for a coffin, and family car for a hearse.
You bring your friend to a nice place, maybe among some flowers, maybe under an old tree, and you start to feel a little better, because you know that they are too.