This is one of the first Diplomatic Policy Cards in the game, unlocking at the same time as the first Governments with Diplomatic slots. Unlike the other Diplomatic Policy, Charismatic Leader, it has a more limited use - it will work effectively only when you send your very first Envoy to a City-state, by simply giving you another free Envoy to the same state. We can't say that it is only useful in the early, or middle game, since you may keep meeting City-States well into the late game (especially if you're behind on your exploration); however it is generally more useful in the early stages of the game, when you haven't yet had many opportunities to influence City-States.
The best way to use Diplomatic League is to preserve your accumulated Envoys until you have the chance to change Governments, then slot this and send one Envoy each in Cities you haven't influenced yet. For the rest of the time, you should have Charismatic Leader in your Diplomatic slot. Of course, this strategy has the obvious flaw that you forego the gameplay benefits of influencing city-states as soon as you are able - you will simply have to decide on-the-go. It should be noted this effect only works on city states you meet after enacting this policy, and will not work retroactively.