Power increase causing voltage drop
You said yourself voltage is like pressure and current like flow rate.  If you measure the pressure in your water line just before the sink valve, it will be maximum when the valve is off, and go down as the valve is opened more (letting more water flow).  The pipes back to wherever the water pressure is maintained have some resistance.  This resistance times the flow causes a pressure drop.

In the case of the processor and its power supply, the power supply is more like a pump.  This is like a water pump that has a maximum flow it can sustain.  If you tried to draw 10 gal/min from a 5 gal/min water pump, you're not going to get a lot of pressure, and you're not going to get 10 gal/min.

It's the same thing with a processor and its power supply.  Let's say the power supply is rated for 3.3 V and 200 mA.  That means it only promises to maintain 3.3 V if you draw 200 mA or less.  If you try to draw more, various things can happen, but they all include dropping the voltage.

Note that the section you quote talks about the processor malfunctioning when it gets too low a voltage.  This is not the same as getting damaged.  Low voltage to a processor is not going to damage it, but you can't expect a processor designed to run on 3.0 to 3.6 V to work at 2.0 V.  At too low a voltage, transistors can't be turned on fully anymore, which means they can't turn on other transistors fully, etc.  The digital levels become undetermined and take longer to propagate thru the logic.  The processor performs meaningless operations on meaningless values, if it can "operate" at all.  At some points the clocks fail to clock.