Have you read this article? It explains what mechanical advantage is fairly well.
The short version is that the gears amplify the torque because they make each turn of the input gear equal a fraction of a turn on the output gear. So turning one gear 10 turns may only get you 1 turn on the other, which amplifies the torque.
The basic mechanism used for mechanical advantage is the Lever. From the linked article:
It boils down to the fact that when one end of the lever moves a shorter distance in the same time, the force must increase for the power to remain the same. Same with the gears.