Note that the correct term for "global warming" seems to be réchauffement climatique, although of course, changement climatique is also common. Already, we would expect réchauffer from that. My guess is that réchauffer is used because there is a natural cyclic warming and cooling cycle; some occasional warming is to be expected, but the current trends are concerning because the increases are unprecedented and are not part of the natural cycle.

• Why is "se réchauffer" used instead of "se chauffer"?

Using se chauffer would more imply some sort of intentional heating method:

Mes voisins se chauffent au fioul.

Should you want to express that the planet temperature is rising, the verb échauffer would be more appropriate than chauffer:

La terre s’échauffe sous l’effet des gaz à effet de serre.

On the other hand, réchauffer expresses the same planet temperature rise, but it also implies a reversal: before that rise, there was a drop. I agree with Maroon that the fluctuating nature of climate favors the use of this verb, which is the most usual in this context of global warming.

Consistency might also play a role because there is no other verb/word than se refroidir / refroidissement to express the opposite (cooling).

There is no *chauffement nor *froidissement in French.