The other answers here aren't wrong. But they don't contain the (whole) correct answer. Namely:
Dumbledore was scared of Greyback.
This is what causes Dumbledore to plead with Snape. He knew that he was going to die one way or another. He had embraced death and wasn't scared of it. But he was worried that he might die at the hands of another Death Eater (like Greyback) who would cause his death to be drawn-out and traumatic.
"You alone know whether it will harm your soul to help an old man avoid pain and humiliation," said Dumbledore. "I ask this one, great favour to you, Severus, because death is coming for me as surely as the Chudley Cannons will finish bottom of this year's league. I confess I should prefer a quick, painless exit to the protracted and messy affair it will be if, for instance, Greyback is involved - I hear Voldemort has recruited him? Or dear Bellatrix, who likes to play with her food before she eats it."
His tone was light but his blue eyes pierced Snape as they had frequently pierced Harry, as though the soul they discussed was visible to him. At last Snape gave another curt nod.
(Deathly Hallows, Chapter 33, The Prince's Tale).
Note that Dumbledore wants to avoid pain and humiliation and is specifically worried about Greyback and Bellatrix. When he's disarmed on the Astronomy Tower Dumbledore knows that his time must be up. Either Snape will kill him or one of the other Death Eaters will. Bellatrix isn't there to do it but, as he had feared, Greyback is.
"Is that you, Fenrir?" asked Dumbledore.
"That's right," rasped the other. "Pleased to see me, Dumbledore?"
"No, I cannot say that I am..."
Fenrir Greyback grinned, showing pointed teeth. Blood trickled down his chin and he licked his lips slowly, obscenely.
"But you know how much I like kids, Dumbledore."
"Am I now to take it that you are attacking even without the full moon now? This is most unusual...you have developed a taste for human flesh that cannot be satisfied once a month?"
"That's right," said Greyback. "Shocks you, that, doesn't it, Dumbledore? Frightens you?"
"Well, I cannot pretend it does not disgust me a little," said Dumbledore.
And he raised a yellow fingernail and picked at his front teeth, leering at Dumbledore.
"I could do you for afters, Dumbledore..."
(Half-Blood Prince, Chapter 27, The Lightning-Struck Tower).
Dumbledore was brave enough to stand up to Greyback. He wasn't going to betray his fear openly in front of the Death Eaters. But that doesn't mean he wasn't scared. He is unarmed and defenceless. Greyback is demonstrating his very real desire to rip him to pieces. That's not a pretty death, as Dumbledore himself says in the earlier quote. Although he also cared about saving Draco's soul, as Dave Johnson says, he is really speaking out of self-interest here. Snape is his one way out of a grisly, painful death. Yet Snape was unwilling to kill Dumbledore - perhaps out of love/respect for Dumbledore and perhaps because killing someone is simply a big deal (Snape has to summon the will to go through with Dumbledore's request). So Dumbledore pleads with him to keep his word and finish him off - knowing the fate that awaits him if Snape should fail.