So why Russia invaded Ukraine?

But I tell you, Bernd, it's actually very simple. It's REALPOLITIK!
According to realism, the international scene is anarchic (ie. there is no regulation, or a higher power that forces an order on it), and states have to make decisions in the fog of war, in a state of incomplete knowledge. And the fact that they can't know exactly what other states are doing, and why they are doing it, this makes them paranoid.
States are obsessing over two things: security and power. They raise security so they can raise their power, and they raise power to raise their security.
Security is both military and influencing others. Power is the ability to make decisions as we know it from the Theory of Power..., and states try to make sure they make their own decisions, but also they try to make decisions for others. The latter is done via soft power (influences, cultural, economical), various diplomatic tools (partially soft power, but partially hard), and with military might. Again they do this to increase their own security and power.
Realists also believe war is inevitable, and if it comes to that, have to act fast and decisively.
So there are three states:
A b C
'A' and 'C' are major powers, 'b' is a minor. When 'A' and 'C' look at 'b' they'll see how weak she is, and how easily could be taken. Since this is the thing that came to their mind, they suppose the other think the same thing. Then 'A' and 'C' will think it would be quite uncomfortable if the other would extend the influence over 'b', so both start to do just that. First they'll use diplomatic tools, they'll try economic and cultural ways, but the race becomes harder as they try to buy the leadership of 'b', perhaps doing even a coup or two, and at one point one of the side we'll recognize she run out of steam and losing the diplomatic battle, and have no tools left but the last argument of kings. So that side will start a war, trying to occupy, annex 'b' - so she can both preserve her own security by denying the other to acquire 'b', and raise her own power by incorporating 'b'.
'A' is the USA with NATO.
'b' is Ukraine
'C' is Russia.

There are many other considerations. Like from the Ukraine the flight time of ICBMs to Moscow is way shorter than those launched from the US, UK, France, or the seas. Or that Ukraine can offer great amount of foodstuff, or big ports, or any other economic assets. Or that there is in fact a sizeable Russian minority there. Or that historically Ukraine is an "artificial construct". Or anything really.
But all just circumstances that make this particular situation unique. And all just rationalizations. In it's core it's this abstract process above.