Pushing and dragging a box
Try to push the box while holding the handle. Have you managed to move it?
Then try to move the box by holding its handle and pulling it. Has it made any difference? Notice that in both cases putting the box in motion is the most difficult part and moving it along is actually a bit easier.
How it works
Force is a vector quantity. That means that to describe it in full you need not only the magnitude (which would be enough in the case of, e.g., mass), but also the direction and sense. The handle determines the distribution of forces when pushing and pulling. The force you apply to push the box may be expressed as the result of two forces (see the picture), one of which acts downwards, pressing the box to the ground and increasing the friction. In the case of pulling, one of the forces acts upwards, which has the effect of decreasing the pressure and friction.
Either way, putting the box in motion requires more effort than moving it along. When the box is stationary, you have static friction which is bigger than kinetic friction present when surfaces move against each other.
Application of the presented laws may be helpful in moving furniture. While pulling a wardrobe, try to lift it slightly upwards. Sometimes too little friction is unfavourable like in the case of a delivery truck without load which moves upwards on the slippery surface.