Espionage in DA

Your espionage level builds up for each turn you have a spy on an enemy planet. Getting "low" espionage may only take 6 spy-turns (one spy for 6 turns, 6 spies for one turn, or anything in between) but higher levels will take longer. This is much improved from DL, as you can prevent your enemies from finding out all your data by killing off their spies as soon as they appear.

Do these espionage levels apply for those specific planets only or for the entire race?

For the entire race.
On which planet you put your spies, is just important for sabotaging buildings that are important to the AI (e.g. it's better to put a spy on a Technology Capital than on a Xeno lab)
I would advise you to either spend a lot on spies early in the game (because then they're rather cheap and quickly trained) or only once you've got more cash coming in than you need. I feel in the midgame it often drains your economy too much (spending several 100 bc's and it still takes more than 10 turns to train a single spy)