What does the value "Z" in the function table for this inverter stand for?
Interestingly, none of the other answers tell you the main purpose of this inverter.

They are all correct : this is a high impedance state. And as you quote in the question, this device needs a pull-up resistor to work.

So why use a gate that needs an extra component?

Because it's a way of sharing signals, of connecting several signals (possibly in different locations) together, to share a common wire. This has various purposes, including communication in both directions on a single wire, as in the I2C bus (which has a second wire for the clock). Or allowing an unknown number of connections to the signal : you can always plug another one in, allowing hotplug connections.

Consider what happens if you connect 2 conventional logic signals together : if one drives "H" and the other drives "L", they fight each other, the actual voltage can be indeterminate, the stronger driver usually wins, and it's possible to burn the other one out... not good.

But connect 2 of these together (with the required pullup resistor) - if either or both is '0', the value is '0'. If all outputs are 'Z', the value is 'H'. That's it.

It's commonly called a "wired-OR" or sometimes a "wired-AND" configuration - though if all the drivers are inverters as in your case, it's actually a wired-NOR structure. Draw out the truth table to confirm this...