What do the capacitors on IN and OUT of an LM7805 voltage regulator?
I'm afraid you need to review capacitors.


  I know that when capacitors are in serial, you add their values.


When capacitors are in parallel, their values add


  Not in serial manner like I was expecting to see.


Loosely speaking, a capacitor has "infinite" impedance at DC.  So, if the capacitor were in series with the regulator output, there could only be AC current through.  Thus, the load would not have a DC voltage across, only an AC voltage.  This is just the opposite of what we want.

When the capacitor is placed across (in parallel with) the regulator output and ground, the capacitor presents a (hopefully) low impedance for AC current through the capacitor and ground, "shunting" the ripple current around the load thus reducing the AC voltage across the load.

But, for DC, the capacitor is effectively open so the full DC voltage appears across the load.  This is just what we want.