When you hyperventilate your carbon dioxide level goes down, which significantly affects blood pH and other body chemistry (respiratory alkalosis), which might cause muscle spasms, as well as decreasing blood flow to the brain, which makes you feel lightheaded.
Re-breathing in a paper bag restores CO2 levels. While it’s true that very high CO2 levels can cause sleepiness, that doesn’t come into play here.
As the lungs lose some of their carbon dioxide through hyperventilation, the volume difference is made up by slightly higher oxygen levels. Remember that most inhaled and exhaled gas is (inert) nitrogen.
With continued rebreathing you will suffer from both hypoxia, as oxygen is extracted by the lungs & not replenished, and hypercarbia, as CO2 builds up because it has nowhere to go.