The basic idea is always the same: there's a global MODE variable that is changed, and the routine that draws the screen does so based on its value:

function drawScreen() {

if (Game.MODE == 1) {

// draw start screen

...

}

if (Game.MODE == 2) {

// draw options screen

...

}

if (Game.MODE == 3) {

// draw level x

...

}

}

When the player goes to the option screen, you simply change the value of MODE.

There are other ways to implement this, but that's the basic gist.

(Btw, the fact that you use SpriteKit is irrelevant to this question.)