Level-Shifter, a resistor i dont understand
It's not a critical choice.  If you were driving many TTL loads with Iih of 60uA each 1K would still be overkill for the DC level.  There is one more consideration.  The capacitance of the input must be considered.  The Rc choice with this capacitance will determine the rise time.  

What is more important here is why use two transistors to do this.  If you put the input to the base of the second transistor you would only need a beta to 10 to saturate the transistor.  If you use a more reasonable 10k pull-up there is not doubt at all.  The inversion of the bit in the micro is much less expensive than the second transistor.  Engineers must always be concerned with the cost of a product.