In a nutshell, anti-aliasing adds subtle pixel color shading to blocky pixel based images to fool the eye into thinking it is looking at smooth graphics.
Think of a diagonal staircase. Now fill the steps with a bunch of cement to make it a smooth ramp. If the steps are pixels on your screen representing a diagonal line, then anti-aliasing acts as the "filler-cement" to make that diagonal line smooth, removing the staircase-effect.