When I first came across this question a few hours after it was asked, it immediately struck me as a homework question. Please don't take offense to that, but it struck me that way-- it's just quite straightforward on the surface of it, which isn't a bad thing. I thought about it for a minute and came up with a few reasons that the Kievan-Rus' association with the Byzantine Empire was politically, economically, and culturally advantageous. 'Of course!' I arrogantly thought, 'the Third Rome Doctrine! The marriage of Ivan III and Sophia Palaiologina! The power of the Orthodox Christian Church in Russia!' I slammed out a few paragraphs and sat back in admiration at my own righteous work. The self-satisfaction this produced was only momentary though because then I immediately began to assign dates to those ideas and events: fifteenth century, sixteenth century, and seventeenth century* respectively.

* I'll defend this assertion below for those who are eager to point out that the Christianization of the Kievan-Rus' in the eleventh century marked the official birth of Byzantine Christianity in Russia. (See the final paragraph of Kievan-Rus' Period and item three in Imperial Period)

In short, all those dates are several hundred years after the generally accepted end date of the state of the Kievan-Rus' (which is marked by the Mongol sacking of Kiev on 6 December 1240 AD)^([1]). In shorter, that means that they don't answer your question satisfactorily.

The Kievan-Rus' period relationship with the Byzantine Empire was almost entirely one based on economic benefit-- you'll also see some subtle cultural influence as well but the real cultural, and certainly political, benefits of this relationship did not begin to pay dividends until several centuries after the state of Kievan-Rus' had been replaced by the more-modern federation of Russian Princes consolidated under the Grand Principality of Moscovy. The Byzantine Empire began earlier and collapsed later than the Kievan-Rus' federation (or however you like to style it), that means the relations between these two states occurred between 880 and 1240, 360 years. All that said though, the Moscovy princes and their subordinate minor princes and dukes most definitely styled themselves as the inheritors of the Kievan-Rus' state, so therefore, I do think those political and cultural benefits that I mentioned earlier should be taken into consideration as well, I also have an asterisk to defend.

Beginnings of Relations

Interaction between the Byzantine Empire and Kievan-Rus' date back to the legendary figure Rurik, who supposedly ruled from 830-879. In this answer I talk a bit about the reliability of the primary source in which Rurik is attested, it's not the most trustworthy document that's ever been written.

Unfortunately for us, Nestor the Chronicler's Essays on Bygone Times (often translated Primary Chronicle) is the only extant document that we really have. Fortunately for us, however, the areas where Nestor is most likely to have been a bit fast and loose with the facts concerns exactly how, why, and when the Norsemen who eventually came to rule the lands later known as Rus' came in the first place, not necessarily about their relations with the Byzantines. Therefore his references to the Byzantine Empire are far more trustworthy than, say, his pedigree of the Varangians, because they more likely to be disinterested and can be corroborated by Byzantine documentation of the same events.

Here is the first mention of the Byzantines (whom Nestor refers to as Greeks) in the Chronicle:

In the year 6360 [852 AD, Nestor is using a calendar which begins at 0 with the creation of Adam-- i.e. Byzantine Christian dating], the fifteenth of the indiction at the accession of the Emperor Michael, the land of Rus' was first named. We have determined this date from the fact that in the reign of this Emperor Russes attacked Tsar'grad, as is written in the Greek Chronicle.^([2])

So, two things to look at here:

1. "The fifteenth of the indiction..." An indiction is a fifteen-year period that is used to measure various cultural and political phenomena, which was introduced by Byzantine Emperor Flavius Valerius Aurelius Constantinus Augustus, or more commonly-- Constantine the Great. A Rus' chronicler using a Byzantine measurement of time is certainly evidence for some cultural influence.
2. "...as is written in the Greek Chronicle" Nestor is referencing a knowledge of the people of Rus' by the Byzantines as far back as the mid-ninth century.

Nestor goes on to make mention of various raiding campaigns launched by the Kievan-Rus' throughout the ninth and tenth centuries, but the Varangians rulers reverted to type, stopped trying to steal from the Byzantines via raiding, and started trying to trade with them. The Varangians were most probably far more mercantile than once thought-- which is how a group of people might be able to come into a foreign community, adapt to its mores, succeed economically, and establish a kind of oligarchy. This was most probably the true kernel of Varangian rule in Rus' as opposed to some kind of cross-cultural trans-Baltic Sea invitation from the Slavic people as Nestor describes in his Chronicle.