Circuit design advice
I think the question is not why they can be removed, so much as why are they in the circuit to begin with? 

What do you think they do?  

Essentially, your schematic is incorrect.  That's why they can be removed :).  

Pull-up and pull-down resistors are only used in the context of digital logic.  That LED display has no logic gates - it's just a bunch of diodes (that happen to emit light as well).  There is no digital logic, so using pull-up resistors is pointless.

Your ciruit would normally employ current limiting resistors in series with the common cathodes, or even better, in series with the anodes, but there is no reason to have pullups here.  The LEDs simply need current to flow when they should be emitting light, and not flowing when they should be dark, and the transistors will do that job admirably and the pull-ups in this case are totally useless.  They'll do little more than burn milliwatts.  

But, why are pull-up/down resistors used at all? Regardless of whether it is specific to your circuit or not?  There are really only two reasons:


To set a default logic level.  Logic inputs are high impedance, and have a voltage range where they will be in a indeterminate logic level, and can do all sorts of nasty things like oscillate between high and low erratically, which if that logic input is connected to other things inside whatever chip has the logic inputs (which is almost always the case), then that can even result in the entire chip behaving erratically or just doing weird but inconsistent things.  


Unless a chip has its own internal pull-ups, then you need to actually set a default logic level on each input pin to ensure they always have a valid logic high or logic low state, even in the absence of a specific input from elsewhere.

Oh, and pull ups simply pull the gate up to your above ground voltage rail, and pull-downs rather pull the gate down to ground, but its really just a matter of if you want logic HIGH or logic LOW as your default.  ...Well, except when its not.


This brings us to the second reason, which is that logic outputs don't always have the ability to output a logic high and low directly.  In fact, most of the time they can't.  A basic part (albeit not really digital, but a very basic part you might have heard of regardless) that demonstrates this well is any of the basic comparators.  The LM311, LM339/LM393, old stalwarts of, uh, comparison.


Their outputs won't work without a pull-up.  They effectively just have an NPN transistor (or more likely if they're CMOS, something like an n-channel MOSFET) connected to ground, with the collector being the actual comparator output.  

In such a configuration, if the comparator swings 'low', what that really means is the output transistor will turn on, connecting the output to ground.  However, when it swings high, all that happens is the output transistor turns off.  This generates no voltage or really any signal on the output - this simply makes the output high impedance, as if it's not there.  If you actually want the output to swing to, say, 5V for high, and ground for low, all you do is add a pull-up.  That way, when the output is 'high' (high impedance), then it might as well not even be connected, and your output is just a connection to your 5V rail through a resistor.  Nice and simple!

When the comparator goes low, then it will suddenly form a connection to ground, causing the resistor to drop most of the voltage, effectively pulling down the output. 

A mental image that I like to imagine is a bunch of people riding a bus, or a train, the kind with that long rope that anyone can pull on to let the driver know that they're stop is ahead.  No one is going to be able to 'push' on the rope to, say, poke the driver with it, so instead, the rope is put up high and then anyone can simply tug down on it.   Well, most outputs can usually only tug down or push up, not both.  There are definitely plenty of chips that can do both, but they will clearly state this fact, and often it is their primary purpose.  Things like gate drivers or beefier buffers will have what are called 'push-pull' outputs, or might mention totem pole outputs.  All this means is their output has both a PNP/NPN or PMOS/NMOS pair on the output, so it is able to actually force the output voltage high or low on its own.  

You'll also see this called sinking and sourcing current, because this also translates into current flowing into (sinking) or out of (sourcing) an output pin.  So another way to put all this is that many outputs can only sink OR source current, but not both.

This is why there are negative voltage regulators, like the LM7905.  You'd think that you should be able to use an LM7805, since ground is just an arbitrary label, and there is no difference between 5V and ground and ground and -5V, as long as the higher potential is labeled higher (5V -> Ground and Ground -> -5V).  So why not just use an LM7805 and call the ground -5V and the 5V out your ground? 

Because the LM7805 can only source current.  The OUT pin can source current, or have current flow out of it, but is totally unable to sink current, or pull current into it.  But, to regulate a negative voltage, that's exactly what needs to happen! In fact, if you try to use an LM7805 in such an application, it will probably get really hot and maybe explode a little (fun!).

This illustrates my point, they are both linear voltage regulators, the circuits are nearly identical, one regulates 16V to 12V, the other -16V to -12V.  The key difference is the top one can source current, and has a NPN pass element, and the voltage reference is a zener diode (which will only work as a reference in one direction - reverse of a traditional diode).  The bottom circuit has a PNP pass element, and the zener diode is reoriented so it will function (since it needs its arrow to point to the higher potential, which for a negative regulator, is ground!).



You might want to look here, as well: https://learn.sparkfun.com/tutorials/transistors/applications-i-switches

It's a terrific little intro that describes these differences in more visual terms, and has examples specifically involving digital input/outputs as well as driving LEDs.  You'll note that there is a series resistor, rather than a pull-up.