Hyperventilation alone does not cause you to feel that you're not getting enough oxygen. Rather, it's what causes hyperventilation that does that (thus resulting in hyperventilation.)
The Calm Clinic explains this quite well (while only mildly contradicting your quote):
During periods of intense anxiety, the body is sent into a state of fight or flight, when the brain signals to the body that danger is afoot. When this happens, you automatically start breathing quickly, as this oxygenates your blood and prepares your body to respond to a threat by fighting or fleeing. If the threat that has triggered your fight or flight response (whether real or imagined) persists, you’re likely to continue hyperventilating until you start to experience other unpleasant physical symptoms.
The focus of your question is
"...This makes all of the symptoms of hyperventilation worse." [emphasis mine]
You can hyperventilate by breathing too quickly or too deeply; either way, in people without underlying medical disorders, hyperventilation is usually caused by stress/anxiety. Anxiety makes your heart rate increase, and causes a perception of the need for more air (not the actual hyperventilation). It is often accompanied by some degree of chest tightness, which many people reasonable attribute to a problem with their heart. These things tend to cause more stress, so it's a cycle. The symptoms of hyperventilation are dizziness/lightheadedness, tingling in your hands/feet and around your mouth, and more but less common symptoms.
Under normal circumstances, hyperventilation leads to a period of decreased respiratory rate to allow for arterial blood to build up that critical buffer, HCO3.
Dealing with Anxiety Symptoms: Hyperventilation
(written for laypersons)