The Arduino code defines a byte (eight bits) based on the code written.
This byte is then passed serially to the shift register.
The shift register then uses the byte to set each of eight pins either high or low (aka serial to parallel conversion). (A useful tutorial using shift registers with Arduinos can be found here
These eight pins are connected to the input pins (four each) on the two 74141 chips.
The 74141 chips read the four bits as a code that defines which of the numbers to light on the Nixie tube. See the datasheet
for the codes)
A Nixie tube works by having a high voltage (typically 150 - 180 volts) attached to the anode. Each of the filaments is connected to the anode and each has a separate cathode. When a number's cathode is connected to ground, current flows through the digit and it lights up).
The 74141 chip is designed to interpret the four bit code to connect one of its ten pins to ground. The 10 cathodes of the Nixie tube are connected to these pins. When one of these pins gets connected to ground, that number lights up. The 74141 is specially designed to handle the Nixie's high voltages.The same thing could be done with a series of transistors, but the 74141 chip just simplifies things.