You have to realise something. Putin is an "ethno-nationalist". For him, Russia is not defined only by borders. It is defined by PEOPLE.
Putin sees himself as responsible for the Russian people - WHEREVER they may be.
You may have heard of that often mischaracterised quote from Putin, where he said the collapse of the Soviet Union was “the greatest geopolitical catastrophe of the century". Most people do not include the more important part of the quote, where he explains WHY that is true: "Tens of millions of our fellow citizens and countrymen found themselves beyond the fringes of Russian territory".
Obviously, Ukraine is a prime example of this, given so many Russians went there to work in the nuclear industry, aerospace, and of course extraction and other heavy industries for which the Donbas was famous.
Putin feels he has a responsibility to protect those "tens of millions" of ethnic Russians who, through no fault of their own, suddenly found themselves "strangers in a strange land" after the fall of the USSR.
These disinherited Russians need and DESERVE the protection of Mother Russia, Putin believes.
That is what we saw on display in Georgia and in Ukraine.
Westerners are far too eager to dismiss Putin's claims of moral imperative regarding these displaced Russian populations. they trot out that old hackneyed "Hitler" metaphor, and try to compare the Donbas with the Sudetenland. This is RIDICULOUS. The Sudeten Germans moved to that area in the 1300's, when it was known as The Kingdom of Bohemia and ruled by the Holy Roman Emperor. They had NEVER been "German" citizens - there was no "Germany" when they settled in Bohemia.
The Donbas Russians, on the other hand, had held Russian passports just a couple of decades ago - less than a generation separated them from their homeland.
In Georgia, the situation is similar although it was complicated by the presence of peacekeepers.