I think you should just play around with it.
Espionage Points (EPs) come from your palace, certain buildings & wonders, spy specialists, and the Espionage slider.
EPs are used passively and actively. Passively, they reveal what your opponents are researching if you have EP superiority over that opponent. Actively, they are used by spy units to perform missions in enemy cities.
Spies require the Alphabet tech, and missions are fairly pricey in terms of EPs spent.
Try adjusting the EP "weighting" system on the Espionage screen. Build a spy and send it into an enemy city and see what your options are.
Personally, I like to use the counterespionage mission when I am in war (or about to be), becuase it greatly reduces the success rate of AI spy missions, like the irritating poisoned water mission. I will also use a spy to create civil disorder, which causes the city to enter unrest (and removes cultural defenses) for one turn, though this is an expensive mission.
Also note that a Great Spy can be used to generate EPs against one particular opponent, which can be very handy if you know you're going to be at war with them or just want to steal technologies. Great Spies can also be setlted as super specialists (for research and EPs) or to build the Scotland Yard, an Academy-like building that improves EP production in the city.