The language of the world is the interconnectedness that exists in all things. Humans, animals, plants, and even inanimate objects are all intricately woven together because everything is a part of the single whole. Once a person believes in that concept, that person can then begin communicating with the universe and individual parts of the universe. This is what Santiago realizes when he starts to be able to communicate with the desert and the wind.
The alchemist is the character that I believe tried to explain it to Santiago the best. He stressed that it was important for Santiago to develop a relationship with nature.
You don’t even have to understand the desert: all you have to do is contemplate a simple grain of sand, and you will see in it all the marvels of creation ... Listen to your heart. It knows all things, because it came from the Soul of the World.
Based on that quote, a reader should realize that it is the heart that understands and knows the language of the world because the heart came from nature and the Soul of the World; therefore, Santiago, the Soul of the World, and the world itself are parts of the whole that can speak and learn from each other.
This heart/language/world bond also applies to relationships, and it explains why Santiago is able to instantly know his connection to Fatima.
It was the pure Language of the World ... What the boy felt at that moment was that he was in the presence of the only woman in his life, and that, with no need for words, she recognized the same thing.
Notice that the quote emphasizes that there was no need for words, just as Santiago communicated with the desert without words. By understanding the language of the world, Santiago gains a great deal of understanding about the world around him. That knowledge grants him power as well as his eventual treasure of wealth and love.