Why does the whole planet Earth have a capacitance?

Because every isolated conductor has one, and Earth counts. Capacitance is the ability of an object to store electric charge, measured as the ratio between the charge it gains and the change in its electric potential. Add charge to a lone conductor and its potential rises, so that ratio exists even with no second plate in sight: it is called self-capacitance. For a conducting sphere in free space, it follows a simple formula, C = 4πε₀R, so it grows with the radius. Treated as an isolated conducting sphere, Earth comes out at about 710 microfarads, while the metal dome of a Van de Graaff generator, typically a sphere 20 centimeters in radius, has only about 22.24 picofarads.

The first device built to store charge came long before the word capacitance. In October 1745, the German cleric Ewald Georg von Kleist tried to accumulate electricity in a small medicine bottle filled with alcohol, with a nail stuck through the cork. When he accidentally touched the nail while still cradling the bottle in his other hand, he received a significant shock. His discovery, made independently by Pieter van Musschenbroek in Leiden, became the Leyden jar, the original form of the capacitor.

Today capacitance is counted in farads, named after the English physicist Michael Faraday: a one-farad capacitor charged with one coulomb shows a potential difference of one volt between its plates. In circuits, what matters most is mutual capacitance, measured between two components, as in a parallel-plate capacitor: two conductive plates insulated from each other, usually with a dielectric material in between. Its capacitance is very nearly proportional to the plates' area and inversely proportional to the distance between them, and for many dielectrics it does not depend on the voltage or the charge.