They are on good terms with us, but not formal allies, which leads to:
The Russians assessed, correctly, that we would not intervene (militarily, at least) on the Georgians' behalf.
The Georgians knew this, which is why they had tried so hard in the proceeding year to get into NATO.
There's also the old Cold War rule that as soon as one superpower deployed forces to a given conflict, the other superpower most certainly would not. That's why the US was so quick off the mark to get involved in Korea and Vietnam, while the USSR was so quick off the mark to get involved in Afghanistan. If you were the first superpower (to be blunt about it, the first nuclear power) to get involved in a war, you would be the only superpower directly involved in that war.
To this day, the US is willing to deploy troops for lots of reasons--sometimes to protect allies, sometimes in response to humanitarian issues (Yugoslavia)--but never in a situation where they will get into a shooting war with Russia or China. And the tendency is reciprocal. When Russia sends troops into Georgia, they do so knowing the US will not send their own troops to Georgia. When the US sails a half dozen aircraft carriers through the Taiwan Straits, as they did in 1996, they do so knowing China will not cross that line and attack Taiwan, as they were threatening to do at the time.
Excellent point. Though I believe that ties into the parent posters point of Georgia trying to get into NATO. If they had been NATO the US would have been legally obligated to help, which would have pushed Russian to avoid invading knowing it would start a much bigger war. Since Russia knew Georgia wasn't NATO, and it was up to the US to decide to engage or not, and the US would avoid having a war with Russia, Georgia was open for invasion.
It actually goes even deeper than that. Georgia pushed for "fast track" membership in NATO and got shut down, hard. The U.S. wanted it, but most of the other members were so set against it that the U.S. backed off and Georgia didn't get it. When NATO denied Georgia membership, they might as well have sent a letter to the Kremlin with the following text:
"Dear Vladimir and Dmitri,
We are absolutely unwilling to risk war in defense of Georgia.
Sincerely,
NATO."
The great irony is that, by taking steps intended to stave off a Russian invasion, the Georgians instead opened the door for one. If they had instead played a slower and more cautious game, they might have maintained sufficient ambiguity and doubt to keep the Russians out. Maybe. Or not. I'm also playing Monday morning quarterback here, so I can't really blame them for trying.
If the Georgians knew that the US would not help them against the Russians, then why did they attack? They got really drunk one night in the general staff room?
They were pretty much between a rock and a hard place. If they failed to attack, their national sovereignty and unity were at stake. They also thought (mistakenly) that they would be able to keep the Russians from interfering by blocking the tunnel between North Ossetia (part of Russia) and South Ossetia (nominally part of Georgia). Based on the (incomplete/incorrect) information they had, this was a reasonable assumption to make.
What they didn't know was that the Russians had already moved several brigades through the tunnel and controlled both sides of it before the fighting in South Ossetia started. But, hey, maybe it really was pure coincidence that those two things happened in that order, in rapid succession.
They are on good terms with us, but not formal allies, which leads to: The Russians assessed, correctly, that we would not intervene (militarily, at least) on the Georgians' behalf.
The Georgians knew this, which is why they had tried so hard in the proceeding year to get into NATO.