You must use a second camera childed to the weapon to show the zoomed image (set Field of View to a small angle - 15 to 30 - to zoom).
If you have Pro, you can assign a RenderTexture to this camera's targetTexture property, what will make the second camera render to this texture. The RenderTexture can be assigned to a new material, and this material used to "paint" the scope (that's probably the way it's done in the linked video).
If you don't have Pro, you can set the second camera's pixelRect property to a small square in the center of the screen (set the camera's depth to 1 to draw over the main camera) and draw the scope body over it with GUI.DrawTexture:

In this case, the scope body is a 308x308 PNG texture with transparency (see credit and reference at the page foot), and the center area can hold a camera image of 196x196 pixels. You can bring the weapon to the aim position and enable the camera and texture drawing with something like this:

var texScope: Texture;  // drag the scope image here
var scopeCam: Camera;   // drag the scope camera here
var rCamera = Rect(0, 0, 196, 196);
var rScope = Rect(0, 0, 308, 308);
var enableAim: boolean = false; // set enableAim to true to aim

function OnGUI(){
  if (enableAim){
    rCamera.x = (Screen.width - rCamera.width)/2; // center rCamera
    rCamera.y = (Screen.height - rCamera.height)/2;
    rScope.x = (Screen.width - rScope.width)/2; // center rScope
    rScope.y = (Screen.height - rScope.height)/2;
    scopeCam.pixelRect = rCamera;
    scopeCam.enabled = true;
    GUI.DrawTexture(rScope, texScope, ScaleMode.ScaleToFit, true, 0);
  }
  else {
    scopeCam.enabled = false;
  }
}

function Update(){
  enableAim = Input.GetMouseButton(1); // aim while the right mouse button is pressed
}

CREDIT/REFERENCE: The scope image above was captured from the game EliteSquad (2nd liked video), created by leonardotry, and edited to make transparent the undesired areas. You can download it for testing purposes - (click here to download).