Its easier to understand if you remove color and shades, and just consider a similar black and white image. With this, the screen is just a bunch of dots in a grid. If you have something going diagonally across them, such as a triangle, you have to decide which dots to turn on and which to turn off. This means you end up with a choppy looking pattern:

   ■
  ■■
 ■■■
■■■■

If you can also show gray colors, you can render that line at a higher resolution and average the values of each small block, resulting in gray near the line and a much smoother image. The more averaging you do, the better the end result will be, and this is why 8x looks better than 4x and so forth.

This is known as super sample antialiasing (SSAA). There are other methods to achieve similar results with various benefits and drawbacks:

Temporal antialiasing (TAA) which averages between multiple frames with slightly different camera angles. This one was made well known by Unreal Engine 4.

Another is called Fast Approximate Antialiasing (FXAA), which uses some image recognition-style techniques (a severe simplification) to find the jagged lines and blur them.