How do current limiting resistors work between ICs?
Many digital inputs are already high impedance, so will only consume in the micro -> milliamps range from a direct connection between the output of a digital device (microcontroller for instance) and the input of another device. Some devices like older transistor (not CMOS style) inputs will consume many times more current to get the same job done. Either way, generally, connections between ICs )unless they are some kind of special output/input with low input impedance) do not need series resistors.

Always check datasheets for typical input current. If you are designing a rugged and durable device which may be used by "fools" then it's a good idea to put limiting series resistors on all digital outputs except for high speed communications lines which should be reasonably isolated from fools anyway. 

Digital inputs are by nature high impedance so other than over/negative-voltage, you do not have to protect them too much. The protection needed for these can be done with external clamping diodes, and/or high resistance value resistors (much higher than the output resistors, because the tiny internal clamping protection diodes in CMOS digital devices are often very very low power rated) in series with the input to limit current to the internal diodes if they are present. 

Digital ICs like logic gates can have their outputs connected directly to the inputs of other gates, without the need of resistors. 

The L293 series are very old though, and their inputs are shown as equivalent circuits in the datasheets as transistor base connections. This means they could actually sink quite a lot of current from your output pins - which can also provide similar amounts of current, but this is not good for the output port nor the overall power rating of your MCU depending on the package and thermal design. Looking further at the datasheet though, typical input current consumption for the logic inputs and the EN pins are only 0.2 microamps  and that's with VCC = 7V.

Summary:


The series resistor will limit the current which would already flow between the two ICs. 
Yes, but the input impedance of the destination is already having an effect. If the connection between the two during a logic HIGH only consumes 500uA without a resistor, with 5V, then the resistance can be estimated as R  = V/I , R = 10K Ohms. Adding more resistance will just lower the current, and "add" to the total loop resistance to ground. Depending on what the end device is though, series resistors can act as voltage dividers and may influence the ability for the target to read the level correctly, but this will not be common.
The resistor will lower the current and reduce overall power, you will most likely reduce the power more than waste it. 
referring to my earlier rant in the wall of text above, yes for safer designs it's common to place lowish (330ohm - 1k ohm) value resistors in series with all outputs, and for inputs (especially ADC) to have clamping and basic R-C filters included. The MCU development board reference design provided with Design Spark PCB program shows heavy use of protection on the MCU's output/input/ADC pins.