You simply can't have an effective conversation without a common language, and it's very difficult to promote your agenda without talking. In our case, we're making a more figurative use of language since obviously the conversation itself requires that both parties understand the words being spoken.
Speaking a person's language also implies understanding who you're dealing with. If you know their agenda, their perspective and their background you have a much greater ability to communicate. More importantly, you have a decided advantage in promoting your own agenda as you'll be able to frame it in a way that makes sense to them.
Imagine a meeting where you're trying to sell the group on your project. You know that one influential voice feels strongly about using the xyz technology. You couldn't possibly care less as long as you get the results you're looking for. You just want to get the project approved.
By navigating the conversation to the topic of technologies and suggesting that the xyz approach is really the way to go, you're going to get this person's immediate attention. If you're further able to present her as the best choice to handle that chunk of the show, you'll doubtless have her enthusiastic backing. You're now in a much better position to promote your own agenda because you understood hers and found common ground.
The art of diplomacy is closely linked to persuasion. It's not a requirement, but the smoother you can keep the conversation, the more likely you are to be productive rather than falling into pointless bickering because someone was offended. Once again, by having an in depth understanding of the people involved, you're in the best possible position to create an atmosphere that lends itself to getting things done.
When you go to the time and trouble of learning someone else's language instead of just assuming they should understand your own, it makes an impression. Rather than bursting into the room and demanding that everyone do it your way, you've demonstrated yourself to be a thoughtful and considerate person. Consequently, you've already started building bridges, that first necessary step in spanning the gulf between you.