The second object gives an opposite but equal reaction
if I'm not mistaken this can be seen by walking. When you push your foot against the ground you'll exert a force against it, and the ground exerts an equal force against your foot, but onto the opposite direction, which is what pushes you forward
EDIT: Another example is when you hammer a nail onto the wall
using the hammer, you exert a force onto the nail, and the nail exerts the exact same amount of force onto the wall, which is what makes it dig in
if the force reaction happened in the direction of the hammer, it would cancel out like you said in the op
nvm second example made no sense whatsoever now that I think about it