Actually, CPR is often used to keep blood flowing through your body by oxygenating it and then manually forcing the heart to pump blood through your body. There is still oxygen in an exhalation, which is why you sometimes see in movies that a character exhales into another's mouth while they're underwater. This is basic biophysics, as the means by which alveoli "take in" oxygen is by simple diffusion. They don't absorb oxygen from the pulmonary air, they merely allow blood oxygen levels to equalise with aerial oxygen levels. Logically, every mouthful of air exhaled will contain some amount of oxygen. And for a person who has ceased breathing, that amount may be enough to keep them alive.
The carbon dioxide doesn't really do anything (unless the patient is suffering from respiratory alkalosis, which is most often caused by hyperventilation, or metabolic alkalosis, which is often also accompanied by hyperventilation as a compensatory mechanism, so it's unlikely that they would be administered CPR), it's the oxygen present in the exhalation that helps the patient regain consciousness.
With regards to the dangers of oxygen, hyperoxia is a concern, yes, but it only happens when the patient is administered very high partial pressures of oxygen, and it only seems to cause serious damage on neonatology and long-term hyperbaric medicine. On a case of life or death, initiating treatment with oxygen at a high partial pressure vastly outweighs any risks of hyperoxia.
EDIT: Regarding the story of the child, I wager it went something like this: Child is the victim of some accident that causes unconsciousness and is almost always fatal (strangulation, drowning, suffocation, blood loss). An overworked, underpaid, uncaring paramedic gets called in, sees the child and presumes him dead (due to a very obvious sign, such as cyanosis, deathly pallor, great amounts of blood lost, lowered body temperature, imperceptible respiration, ligature marks around the neck, etc). Not to earn the ire of the family, the paramedic does a token and half-hearted attempt at resuscitation (which fails because it's half-hearted). The paramedic doesn't take the time and effort to take the child's pulse properly (which is notoriously hard to do correctly, even by professionals) and feels up a tendon instead of an artery. The child is pronounced dead and a death certificate is emitted immediately, most likely in situ. The child is quickly taken to the mortuary and prepared for the wake, as it's customary on tropical and sub-tropical countries. The child wakes up on his own after a few hours, asks for water, but quickly succumbs to (making a wild guess here) metabolic acidosis. Asking for water could be a sign of hypovolaemic shock (most often caused by blood loss), which would fit the above situation well.
