Background is in draw() so that basically every new draw() is wiped clean with the background, and nothing from the previous draw()'s will show up. Background doesn't set a background color like you may think, it actually covers the entire sketch in a single color. So background(255,0,0) will cover the entire sketch in red with nothing else, you can do this at the start of your draw and then draw whatever else is on your sketch after background().

So you basically just have to have the three background()'s first in draw with the colors you want, but only have one actually executed and the other two ignored based on which color you're currently on.

1. PImage blackLeft, blackRight, blueLeft, blueRight, rot1, rot2, rot3;
2. int width = 188;
3. int height = 184;
4. int currentImage = 0;
5. 
   
6. void setup() {
7.   size(764,1080);
8.   blackLeft = loadImage("black_arrow_left.gif");
9.   blackRight = loadImage("black_arrow_right.gif");
10.   blueLeft = loadImage("blue_arrow_left.gif");
11.   blueRight = loadImage("blue_arrow_right.gif");
12.   rot1 = loadImage("rot1.png");
13.   rot2 = loadImage("rot2.png");
14.   rot3 = loadImage("rot3.png");
15. }
16. 
    
17. void draw() {
18.   switch(currentImage) {
19.     case 0: 
20.       image(rot1,0,0);
21.       break;
22.     case 1: 
23.       image(rot2,0,0);
24.       break;
25.     case 2: 
26.       image(rot3,0,0);
27.       break;
28.   }
29.   image(blackLeft,100,100);
30.   if (mouseX >= 100 && mouseX <= 100 + width &&
31.       mouseY >= 100 && mouseY <= 100 + height )
32.   {
33.     image(blueLeft,100,100);
34.   }
35.   image(blackRight,300,100);
36.   if (mouseX >= 300 && mouseX <= 300 + width &&
37.       mouseY >= 100 && mouseY <= 100 + height )
38.   {
39.     image(blueRight,300,100);
40.   }
41. }
42. 
    
43. void mousePressed() {
44.   if (mouseX >= 100 && mouseX <= 100 + width &&
45.       mouseY >= 100 && mouseY <= 100 + height ) 
46.   {
47.     currentImage--; // When left button is clicked, decrease currentImage by one
48.     if (currentImage < 0) {currentImage = 2;} // When currentImage is 0 and decreased it will become -1, when this happens set it to 2
49.   }
50.   if (mouseX >= 300 && mouseX <= 300 + width &&
51.       mouseY >= 100 && mouseY <= 100 + height )
52.   {
53.     currentImage++; // When right button is clicked, increase currentImage by one
54.     if (currentImage > 2) {currentImage = 0;} // When currentImage is 2 and increased it will become 3, when this happens set it to 0
55.   }
56. }