In fact, Stellaris followed the same underlying logic as its predecessors, Crusader Kings, Europa Universalis, and Victoria. At heart, it was still an ordinary strategy game dressed up in science fiction. Its wars bore little resemblance to interstellar warfare and instead drew heavily on medieval warfare in feudal Western Europe.
Every war needed a pretext, and Stellaris was no exception. There were no games of intrigue with aliens here, nor any fleet tactics to speak of. Just as in the feudal era, declaring war required a suitable casus belli, after which both sides lined up their fleets in space (cavalry forming ranks on the great plains of Europe) and exchanged fire between warships (knights charging into one another).
You can spend influence to fabricate claims on another country's territory, then declare war on that country using Take Claim as your casus belli. Win, and you receive every system you claimed.
Of course, you can also fight a humiliation war on the grounds of humiliation, lowering the other side's happiness if you win. There is also an ideology war, which lets you impose your own ethics on the other side after victory.
Then there is threat removal against a crisis or a Genocidal Empire. This is a war to the death: neither surrender nor peace negotiations are possible, and it only ends when one side is completely destroyed.
Against a Crisis Empire, this casus belli is called existence eradication.
But keep this in mind: you have to occupy all of the opposing country's territory before it will surrender. Otherwise, all you do is rack up war exhaustion for nothing. After the war begins, you therefore need to send your fleets into enemy territory. They will fight automatically, defeating enemy fleets and stellar bases for you.
Fleets cannot occupy colonies directly. You