" just swap CPU and startup windows like it's nothing?"
yep...
What you do need to do is pick a CPU that has a matching socket type as your motherboard.
Example: a 4970K will not run on the same motherboards as a 7700K because they have different socket types.
Other than that it's fairly easy and all you need to do is ensure that the new CPU has the fan, heatsync and thermal paste attached properly.
Check your temps to ensure it's properly attached.
Last note is that some CPU's have abilities that others don't. Often if it's a minor CPU upgrade to the same socket type you'd see little to no difference. But upgrading from an i3 to an i7 might warrant a re-installation of the OS due to the increased CPU ability.
These CPU features I'm referring too aren't just more cores and threads but something Intel refers to as "Advanced Technologies".
Example:
Intel Core i3 6100 doesn't have Intel TSX-NI but the 6600K DOES. Since it's a change to the possible instruction sets for the CPU a re-installation of windows is best because I'm not certain windows would expect or look for those added abilities. That's why, as Lutfij said it's good practice to re-install after a CPU swap.... Required? No.. Good idea? Yes..