Anti-aliasing is difficult to explain. It's easier to show IRL to someone. Like most things its more significant in-motion than in static .jpg images.

For an example of the difference anti-aliasing can make on image quality, try lowering your in-game resolution to one half of what you normally play as. Then go back to the resolution you normally use. That's, in a nutshell, the difference between anti-aliasing and no anti-aliasing. Of course, like most things in life, anti-aliasing is subject to diminishing returns and while it may not be as significant a difference in overall appearance as halving your resolution it works on the same principle. Anti-aliasing works ((SSAA, the good kind) by rending the game's image in 2x, 4x, 8x, whatever your card can handle and then displaying it in your monitor's display size. It's generally also much more performance friendly than forcing your graphics card to actually run at that multiplied resolution and consider the fact that they don't make monitors which can support them anyway. SSAA gives you an image quality that is not otherwise possible to achieve.