It is not always the small space that triggers these emotions, but it is more the fear of the possibilities of what could happen while confined to that area. When anxiety levels start to reach a certain ppo level, the person may start to experience:

sweating and/or chills
accelerated heart rate and a rise in blood pressure
dizziness, fainting spells, lightheadedness and frozen in fear
dry mouth
hyperventilation
hot flashes
shaking or trembling and a sense of "butterflies" in the stomach
nausea
headache
numbness
a choking sensation
tightness in the chest/chest pain and difficulty breathing
an urge to use the bathroom
confusion or disorientation
fear of harm or illness

Causes
The fear of enclosed spaces is an irrational fear. Most claustrophobic people who find themselves in a room without windows consciously know that they are not in danger, yet these same people will be afraid, possibly terrified to the point of incapacitation, and many do not know why. Amygdala
The amygdala is one of the smallest structures in the brain, but also one of the most powerful. The amygdala is needed for the conditioning of fear, or the creation of a fight-or-flight response. A fight-or-flight response is created when a stimulus is associated with a grievous situation. Cheng believes that a phobias roots are in this fight-or-flight response. In generating a fight-or-flight response, the amygdala acts in the following way: 
The amygdalas anterior nuclei associated with fear each other. Nuclei send out impulses to other nuclei, which influence respiratory rate, physical arousal, the release of adrenaline, blood pressure, heart rate, behavioral fear response, and defensive responses, which may include freezing up.
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