When we push a box, the box applies an opposite force on my hand, but why does my hand move with the box as I push the box if the net force of is 0? When we push a box, the box applies an opposite force on my hand, but why does my hand move with the box as I push the box if the net force of my hand is 0? What is happening?
 A: This ambiguity is completely cleared if you try looking with free body diagrams(FBD).
Your hand is a continuous mass distribution if you try looking at FBD of any section you would see that the section experiences two forces on  sides  ---

*

*Force from third law as a contact or normal force by the part of hand pushing against the box.

*Force applied by part of hand  that is transmitting the muscular force

and if you are able are able to push the box that is because for every cross-section of your hand the second force is greater. Every section does receive the reaction force  but it also experiences a force that is being transmitted by muscles.
A: 
Why does my hand move with the box as I push the box if the net force of my hand is $0$?

No , the net force on your hand is not $0$. It is true that the box is exerting a reaction force equal in magnitude to the applied one on your hand but while considering the motion of your hand you have to consider all the forces on it which will be evident when you draw the free body diagram exclusively of your hand.
