Tom,
... how does that charge come to be on its plates? ...
Charges are everywhere around us. Materials are made of atoms full with electrons.
The electric charges around us may have very different "mobilities".
Deep electrons in atoms may need an x-ray or a gamma-ray to be separated from their atom.
Free electrons in metals (also called conduction electrons), on the contrary, are very free to move from one atom in the metal to another. They could even leave the metal with a little energy, but the positive charge they leave behind will attract them back if the energy is too low. Even non-metallic substances can leave electrons rather easily, like http://www.du.edu/~etuttle/electron/elect27.htm" . This is also how thermocouples create a small but measurable voltage when heated: the two metals in the junction have different attractiveness for electrons and electron choose their side.
Besides, there are many different ways to drive electric charges into motion.
The oldest known method is friction (see
http://www.amasci.com/emotor/vdg.html").
The most used drive is the electromotive force created by magnetic induction.
Radioactivity could be used since energetic radiation can quick off electrons (the photomultiplier for example)
Chemistry offers many other possibilities. Electrons may have more affinity for one substance and will move if they are given to opportunity to do so. (this is how batteries work).
In the end, it may happen that the preferred location for electrons are on the plates on your capacitor. They may prefer to leave the battery, at least as far as this is possible: an electrical equilibrium will stop the flow at some time (when the capacitor is charged at the potential of the battery).
So, charges are no mystery at all. Like anything here on Earth it can move, it can be moved, can be manipulated.
Michel