This goes way back to the Varangian guard.

The varangian guard was a mercenary military order (but you can think of it as a french foreign legion sort of thing) that was established, maintained and paid for by the Byzantines. The people joining this guard were from mostly slavic and germanic lands. Now some of those people, upon arrival in Byzantium, were pagans, but over the course of their service, many would switch and become Christians.

At some point, due the large number of christian varangian veterans, the byzantine emperor saw it fit to permit these veterans to establish themselves outside of the borders of the Empire, under imperial protection. The most prominent such establishments were in what is today Ukraine, where varangian veterans became local lords, chieftains or even kings. As mentioned, these people were christians (orthodox). But they brought with them a few more things. One of them is writing. Slavic alphabet was created in greece basically in the late IXth century and as veteran varangians took hold of what is today Ukrainian lands, they brought writing with them. The other is icons. Iconongraphy was booming and a lot of beautiful orthodox icons inspired and helped convert the local population, from whatever pagan religion they had to orthodoxy.

This trend spread and very soon, all the slavic world became orthodox.

As /u/SamuraiRake mentioned, at that time, islam was a very big contender too, but it was the nomadic hordes that took up islam while the slavs took up Christianity. This is for a variety of reasons. There is a comical parable which the aforementioned user also said, that the russians upon hearing that muslims don't drink, they went with Christianity.. but in reality the die was cast long before that.

So to add up, varangians brought contacts from Byzantium to the slavic people. They also brought orthodoxy, icons and writing. Icons were great for impressing and converting the illiterate population while writing was good to solidify and consolidate the religion.

Vladimir the Great of Kiev became Orthodox Christian in the late 10th century as he saw the beauty and prosperity of the Byzantine churches. He found Judaism uninteresting and cited that they had been abandoned by god, as evident by their loss of Jerusalem. He disregarded Catholicism as the German churches didn't interest him in comparison to the grandiose and remarkable Byzantine/Orthodox churches. Finally, Islam was the only real [regional] competition, but he decided against it since it's prohibitions on alcohol and pork, he's even been quoting to saying "Drinking is the joy of all Rus'"

EDIT: Rainman has a fantastic answer, better than mine. I completely forgot about the Varangian Guard, but I did feel like I was missing something.