I recently corrected (increased) the sizing on the capacitors around a voltage regulator and then realized I didn't really have good intuition on what it would look like in the wild if they were wrong.
What is the affect or symptoms I would see on the output voltage if the voltage regulator caps are undersized?
From the datasheet, the first input cap is simply 'required' with no explanation as to why.
*Required if regulator input is more than 4 inches from input filter capacitor (or if no input filter capacitor is used). **Optional for improved transient response.
For the optional second capacitor, I usually choose a nominal value and type as given by the datasheet but don't really have a rule-of-thumb formula to increase that value if, say, the destination is very far away (e.g. microcontroller power pin).
Understanding "Stability" in regulators
To get an intuitive understanding (which is what the question is asking for), you need to understand the concept of stability and how regulators work in the general case.
For most capacitor values given for regulators, the values given are the minimum value needed for stability plus a little margin.
The regulator is a closed-loop system. It watches what happens on the output and adjusts "stuff" internally to make sure the output (really a scaled-down version of the output) always equals a desired value.
Problems occur when it starts chasing its tail. If, as a result of it changing "stuff" internally, the input voltage also starts to change (or the output changes too quickly) then the changes the regulator made will have too much of an effect and it will have to undo the excess.
This corrective change can also overshoot the mark, requiring another corrective change... as you can see, without sufficient "stability" in the system, the regulator can output a continuously fluctuating voltage rather than the flat line you hope for when employing a regulator.
The capacitors slow down voltage changes, thereby helping to ensure overall stability.
The input capacitor is required to stabilize the input voltage. If the input voltage is isolated from the power source by a large inductance (like a long wire) then current changes in the regulator will manifest as large voltage changes at the input due to the inductance. The capacitor is there to "cancel" that inductance and ensure a slowly varying (e.g. stable) voltage at the input. This makes the input change slower than the reaction time of the control loop of the regulator -- achieving stability.
If you don't have a large enough output capacitor (the second one in your description) then the output ripple will be greater than the datasheet predicts. That means you will get the regulated voltage but the output will fluctuate around that voltage. Specifically, the output will be slow to react to rapid changes in load (current demand). So when you activate different parts of your load (application) circuitry you may experience sudden drops in voltage. Those drops can be severe enough to trigger resets in your digital circuitry (or worse, latching) or conditions in your analog circuitry were there is a recovery time (such as discharging filter capacitors).
Rule(s) of Thumb(s)
These are approximations. The correct answer in engineering is always "it depends," but that's not useful. So here I give you some thoughts on "guessing" reasonable values.