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. This doesn't mean that it's useful only 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 to each city-state 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 works only on city-states you meet after enacting this policy, and will not work retroactively.
The ancient Greeks were enamored of diplomatic leagues – loose and limited alliances between city-states, cobbled together for a variety of purposes. In the Archaic Period, the Amphictyonic League was a religious association of tribes. By the end of the 7th Century BC, Sparta was the most powerful city-state in the Peloponnese; to counter its influence some 150+ cities formed the Delian League in 477 BC under the leadership of Athens. It spent most of its existence fighting off Persian invasions. Around 338 BC, Philip II of Macedonia formed the Hellenic League … and Alexander conquered (briefly) the known world with it. The Achaean League existed between 280 and 146 BC, formed to resist the Romans. Didn’t help.