Why can we assume a current direction when using kirchhoff's circuit law? 
Reference direction: 
  When analyzing electrical circuits, the actual direction of current through a specific circuit element is usually unknown. Consequently, each circuit element is assigned a current variable with an arbitrarily chosen reference direction. When the circuit is solved, the circuit element currents may have positive or negative values. A negative value means that the actual direction of current through that circuit element is opposite that of the chosen reference direction.  

Why can we assume a  direction and get the correct value and sign? Is there a simple proof of this fact?
 A: 
Is there a simple proof of this fact?

Sure; when you insert an ammeter into a circuit branch, there are two choices of polarity which amount to choosing a reference direction.
If two identical ammeters, connected in series with opposite polarity, are inserted into a circuit branch, they will measure the same current but give the opposite sign since each has a different reference direction.
However, they both give the same information. For one ammeter, current enters the positive lead and this ammeter gives a positive reading. For the other ammeter, current exits the positive lead and this ammeter gives a negative reading.
In either case, the ammeter reading gives you the correct direction of the current.
A: You can assume arbitrary but then fixed current directions in the circuit branches of a circuit because the result just changes sign when you change the chosen direction. Thus if you change in one branch $n$ the assumed positive current direction your calculation result for the current $I_n$ changes to -$I_n$ so that the actual physical current value and orientation stays the same. It is similar to the case when you consider a body with velocity $\vec v$ in positive x-direction and then make the coordinate transform $x \to -x$. The body still moves in the same direction but the velocity is $-v$.
Let's assume you have a one loop circuit, a battery and a resistor connected to it. Now you chose an arbitrary direction (draw an arrow) in the resistor and say that a current flowing in this arrows direction will be called a positive current. If you draw your arrow going from the plus to the minus terminal of the battery, you will have a positive current, because charge flows through the resistor from plus to minus. If you draw your arrow in the opposite direction (from minus to plus) you will have a negative current in that direction meaning that you have a positive current from plus to minus as before when you described the same situation with the arrow from plus to minus.
