How does the MAX232 double the voltage?
It uses one charge pump to double the supply voltage, and the second charge pump to invert it.  The idea behind the charge pump doubler is that capacitors are first charged in parallel, then they are switched such that they are connected in series.


(Source of picture: datasheet for ICL232, which is similar to MAX232.)

As an aside, I've seen hacks where +10V and -10V generated by the MAX232 were also used as supply rails for OpAmps.  It's not the best power supply, and it's got switching noise from the charge pump.  But it may still work, if the analog section is not very sensitive, and it needs a negative supply rail, and there is no other option for generating the negative supply rail.