The speaker of this line is Mourad. He is speaking to his cousin Aram. Mourad has stolen a beautiful white horse and has asked Aram to join him on a ride. The two of them ride off for a while and enjoy their newfound freedom. Eventually Mourad asks Aram to get off the horse, so that Mourad can ride alone for a bit. Aram responds by asking if he can ride the horse alone too. Then Mourad responds in an odd manner. Mourad says that the horse will decide who rides alone on it. Mourad says that he has a way with horses, and that is why the horse lets him ride. Aram takes slight offense and says that whatever special ability Mourad has, Aram has too. Mourad responds with the quote in question. The full exchange is as follows:
"We rode and my cousin Mourad sang. For all anybody knew we were still in the old country where, at least according to some of our neighbors, we belonged. We let the horse run as long as it felt like running.
At last my cousin Mourad said, Get down. I want to ride alone.
Will you let me ride alone? I said.
That is up to the horse, my cousin said. Get down.
The horse will let me ride, I said.
We shall see, he said. Don't forget that I have a way with a horse.
Well, I said, any way you have with a horse, I have also.
For the sake of your safety, he said, let us hope so. Get down.
All right, I said, but remember you've got to let me try to ride alone."
What Mourad means is that the horse lets him ride without any problems, so he is safe. Mourad hopes that the horse lets Aram have a safe ride. If the horse doesn't treat Aram with the same "respect," then it could be a dangerous ride for Aram. He may get bucked or thrown. That is precisely what happens, too. Aram tries to ride alone and is eventually thrown off. His ride was not as safe as Mourad's ride.