You have to understand that the current through an inductor will increase steadily if there is a positive voltage across it and decrease steadily if there is a negative voltage. The bigger the voltage, the faster the current changes.
When the MAX1771 turns on the MOSFET, there is 6 V across the inductor, so the current increases. The current is flowing from the supply, through the inductor, through the transistor, and through the resistor to ground. The purpose of this is only to build up current in the inductor.
When the current is large enough, the voltage across the resistor gets up to 0.1 V, the MAX1771 turns off.
The current in the inductor cannot change instantaneously, so it carries on flowing from the 6 V supply, through the inductor and the diode and onto the output. So there is never any current flowing to the output at the same time as there is current in the resistor.
Because there is a lot more voltage across the inductor (172 - 6 = 166 V) as opposed to 6 V, the current in the inductor falls much faster than it increased. So although the peak current out is the same as the peak current in, the average current is about 166/6 = 28 (ish) times smaller than the average input current.
You should allow for a peak current in the inductor of about 4 amps.
However, I would question how much current your Nixie tubes need. This big Nixie Clock
https://www.kosbo.com/index.php?option=com_content&view=article&id=53&Itemid=61 takes less than 4 W.
If you really do need 12 W or so, why are you going for a 6V supply? I would suggest a larger supply voltage if you do need that much power.