You are confused because the Vg voltage COMPARED TO "ground" (or the bottom, negative power supply rail) is zero, but compared to the source pin, it is actually negative few volts (Vgs = -x volts), and a P-channel MOSFET conducts or is turned on when the gate pin is a negative few volts (usually around -3V to -10V).
The text mentions the gate "voltage" (the correct term here would be "potential"), but it refers to it as relative to the ground (or the negative power supply rail) instead of to the MOSFETs source electrode, and that's where all the confusion is coming from.
It's really not your fault, but the fault of the person explaining it. It is possible that the author of that schematic and the related text/explanation doesn't understand it well him/herself.
Your schematic symbol for PMOS is flipped around - the arrow is supposed to be connecting on the Source side (to the left).
It would help you to use an NMOS transistor as an example to understand the switching action.
As you probably know, when the gate pin of an NMOS (N-channel MOSFET) transistor is positive compared to the source pin (also known as Vgs or gate-to-source voltage), the transistor will start conducting (a current will start flowing from drain pin to source pin).
Normally you need a few volts for Vgs to turn a MOSFET on, most often around 10V and 5V for logic-level MOSFETs, even less for special types, but usually never more than 20V as that would damage most MOSFET gates.
If you drop the voltage between gate and source to zero (Vgs=0V), the transistor will not conduct, it will be off.
Now all you need to do is flip the polarities around, and you will understand how a P-channel MOSFET works.
Don't worry if it takes you a few times to digest and understand, it is not always easy to wrap our heads around some things, even if they seem simple once we understand them.