See, the problem here is that you aren't redrawing the canvas. In order to achieve this effect, quite simply, you will have to redraw the canvas. There are a number of ways that you can do this, and I see that you are (probably) a beginner. You could use a PGraphics or manipulate the pixels array. Both are fairly advanced, but using pixels is shorter, so I'll demonstrate with that:

In this case, translate() is only useful if you are redrawing everything, but with the pixels approach, you are simply moving everything one pixel to the left every frame.

Revised Code:

1. void setup() {
2.   size(600, 200);
3.   background(212);
4.   smooth();
5. }
6. 
   
7. void draw() {
8.   stroke(#000000);
9.   strokeWeight(4);
10.  
11.   if(mousePressed) {
12.     line(pmouseX, pmouseY, mouseX, mouseY);
13.   }
14.  
15.   loadPixels(); //Prepare the pixels array
16.   for(int i = 0; i < pixels.length; i ++) { //Loop through all of the pixels, position 0 is the top left corner going right and then down
17.     if(i % width != 0 && i < pixels.length - 1) pixels[i] = pixels[i + 1]; //Assign each pixel to be the pixel to its right...
18.     else pixels[i] = color(212); //...unless it is at the end of a line or is the last one, in which case we make it the background color
19.   }
20.   updatePixels(); //Update the pixels array to the screen
21. }