How did Dumbledore know that Harry would believe Severus?

Your question is:
Why did Dumbledore think Harry would believe Severus?

Authentication is always crucial while relaying critical information; and as prats110892 notes, this, given Snape's cunning and access to the pensieve and Dumbledore's portrait together with Harry (once he found him), was not impossible.

Another motivation for Dumbledore's judgement lies in one of his good guesses/speculations that Harry himself suspected that the bond between him and Voldemort was deepening. We see that Dumbledore - later in the same conversation as in the question - mentions (Emphasis mine):

“Meanwhile, the connection between them grows ever stronger, a parasitic growth: Sometimes I have thought he suspects it himself. If I know him, he will have arranged matters so that when he does set out to meet his death, it will truly mean the end of Voldemort.”

- Harry Potter and the Deathly Hallows, Ch. 33: The Prince's Tale

It may as well be that he trusted Harry to realize that the connection was much more than a mere telepathy. He claims to know Harry, after all. So even though, as E. J.'s answer already mentions (and is quite evident from this chapter), Dumbledore trusted Snape to competently carry out this task as he had done countless others, in the end he rested all his hopes on Harry to make the 'connection' once Snape had, by any means, succeeded in getting the message across; and acknowledge the truth about the bond, given all the effects of the scar that he had experienced and use it to defeat Voldemort.

After all, he arranged Harry's Occlumency lessons with Snape, of all people, and expected Harry (albeit a little too much) to not mind that and realize the underlying importance of mastering the skill to keep Voldemort at bay. I sense a parallel here as well.

(The only canon answer I could find to the question, and a first for me too :D)