> ...but once the image has been taken, the exposure has already happened. 
> How can you change it after the fact?

Think about how the camera works (in a very simple way).  It counts photons 
for a certain length of time, applies a scale factor, then gamma correction 
and writes the byte values to a JPEG file.

If you start with the JPEG file, you can undo the gamma, scale the data 
using some exposure-adjustment factor, then reapply the gamma.  You then 
have a new file that will look similar to if the camera had used a different 
exposure.  Obviously the further your scale factor is from 1.0, the more 
artifacts will be introduced to the image.

However pros use the raw sensor data from the camera and not a JPEG.  This 
allows them some margin to adjust the exposure later without adding any 
artifacts to the final JPEG image they create.  Because of this it is 
extremely important not to saturate the sensor (ie 100% white) in any areas, 
it is impossible to get back detail in areas that are at 100%.  If you 
under-expose it you can scale up the brightness without introducing 
artifacts (because usually the raw sensor data is higher bit depth than 
JPEG).

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