As we learn of more and more contacts between members of Trump's team and Russian officials or go-betweens, both in the United States and in Moscow, let's engage here in a few theoretical questions as to what might have been going on. And we don't need to get too conspiratorial about this; at this point, the simplest explanations are still likely the best.
And the simplest explanation for Russia's peculiar attentions towards Donald Trump's top advisers is the one we already know. From the declassified version of the intelligence community's report on Russian state espionage efforts during the 2016 election:
Russia’s goals were to undermine public faith in the US democratic process, denigrate Secretary Clinton, and harm her electability and potential presidency. We further assess Putin and the Russian Government developed a clear preference for President-elect Trump. We have high confidence in these judgments. [...]
We assess Putin, his advisers, and the Russian Government developed a clear preference for President-elect Trump over Secretary Clinton.
This "preference" for Trump developed late in the campaign. The original goal of Russian hacking was, according to intelligence efforts, similar to previous Russian efforts in other nations; to sow mistrust of democratic institutions in general and of Russia-damaging news stories in specific. That goal morphed into an effort to support Trump specifically, in the closing months—after, as we're now learning, numerous meetings between Russian government figures or proxies and member of the Donald Trump campaign.
Let's presume that, no matter how many meetings there were or who they were between, nobody on either side was stupid enough to discuss the Russian hacking efforts that were at that point well underway. There would be no need; the Russians didn't need anything from Team Trump to further their own espionage efforts. And while it's similarly intriguing to think about what-ifs like "did anyone on the Russian side mention to Trump's team that the Republicans had been compromised too?", that particular scenario, blackmail, is also a bit far afield of the the few facts that have trickled out. It's not likely.
So why was Trump's campaign attracting so much Russian attention, during the summer and fall of the election?
In October, Donald Trump Jr. was being paid an "appearance fee" of at least $50,000 to attend a small Paris meeting working towards a Russian-backed Syrian peace plan. Nobody on the planet has ever suggested that Donald Trump Jr. has either an interest or an expertise in Syrian peace efforts, so his appearance was unquestionably due to his status as son of the Republican presidential candidate. The Syrian party leader whose husband arranged the meeting freely stated that she relayed Trump Jr.'s thoughts on the matter back to Moscow.
In September, the Russian ambassador himself—identified by various outlets as tightly connected to the nation's espionage efforts—paid an apparently impromptu visit to Sen. Jeff Sessions, who at the time was the chair of the Trump campaign's national security advisory committee. While Sessions describes a wide-ranging discussion about nothing in particular, he did allow that the subject of Russian military action in Ukraine "somehow came up", and that it was a "testy" conversation. The Russian ambassador did not, seemingly, ask a range of sitting senators their thoughts on the same matter; only Sessions.
The Russian ambassador gets around; he was also at the Republican National Convention back in July, where he met briefly with Sessions and perhaps not-so-briefly with other Trump advisers like J.D. Gordon. This convention was where the Republican party platform stance on the Ukraine conflict was memorably changed to language more favorable to Russia, language J.D. Gordon confirmed he personally "advocated for" on behalf of the Trump campaign, after an earlier meeting between Trump and the (Sessions-led) national security team in Washington. And leaks indicate there have been other meetings, between other campaign officials and Russian proxies; we simply don't know the scope of those yet.
It's the September and October meetings that are intriguing. The Paris meeting was fairly directly an opportunity for pro-Russia voices to pick the mind of a man very, very close to the presidential candidate and to plant ideas of their own; $50,000 for the opportunity is, as any American lobbyist might tell you, chump change for such access. A stumbling Jeff Sessions seems unable to come up with any reason why the Russian ambassador might want to meet with him that September, either in a senate or campaign capacity, other than for small talk—but given that Russian involvement in Ukraine, the single most critical issue in U.S.-Russia relations, "somehow came up", we can glean that the ambassador had an interest in learning what the chair of Donald Trump's national security advisory team thought about the topic, and about the new U.S. sanctions against Russia. (Jeff Sessions might not have been able to suss this out, but it seems rather obvious to the rest of us. Jeff Sessions is either hopelessly naive about these things or is attempting, again, to deflect.)
Each of those conversations was relayed back to Moscow. And, either after those meetings with Trump Jr., with J.D. Gordon, and with Jeff Sessions or concurrent to them, Moscow made a key late decision in their own espionage efforts. No longer would they merely seek to damage Clinton; now, the efforts would be expanded to assisting Trump. We can presume that whatever was discussed during those various meetings, Moscow liked what it heard.
So at the most base level, the attention from the Russian ambassador seems perfectly explainable: Moscow was probing Sen. Jeff Sessions and other campaign officials in an effort to determine whether Trump was worth not just indirect support, via anti-Clinton hacks, but more direct support—say, by withholding similar anti-Trump documents they had obtained from hacking Republican entities, or through the dissemination of pro-Trump "fake news" of the sort that would, by the end of the campaign, be so overwhelming Facebook and other social media sites as to become a major news story of its own. To be sure, the Trump campaign had always been a consistent flatterer of Putin and publicly critical of anti-Russian sanctions—but direct meetings between the let's-say-espionage-familiar Kiyslak and Trump team members would allow Moscow to more directly feel out whether those moves were genuine, or mere campaign fluff.
Kiyslak would have been stupid not to take the opportunity.
That doesn't answer all the questions, of course. The sheer extent with which Trump campaign rhetoric has bent towards Russian goals and needs, during the campaign, has baffled both neutral observers and most of the Republican Party itself. And more then a few events in the timeline line up close enough for observers to wonder whether the quid and the quo have more connection than simple coincidence. But it’s no mystery as to why the Russian ambassador himself would apparently be paying very close and personal attention to what key members of Trump’s campaign team thought about Russia and U.S.-Russia relations.
He was attempting to glean whether or not the members of Trump’s team were worth expanding Russian espionage and propaganda efforts from merely discrediting the U.S. elections to actively supporting one candidate while sabotaging the other. The intelligence community would later determine that Putin decided to do precisely that.