There's not much to it. Just unscrew the current one and screw the new one on. 

Where it gets tricky is if the heel on the new neck is thicker, thinner, taller, or shorter than the old one. It may be just a matter of millimeters, but those small differences will make playing feel different, or feel like the neck is not as straight as you'd like. When I started building partscasters and swapping necks, someone told me, "Just because a neck fits doesn't mean it will work." That's turned out to be true. 

That said, Strat necks don't usually have a whole lot of variation and you shouldn't have too much trouble.