Central to the Sapir-Whorf hypothesis is the idea of linguistic relativity--that distinctions of meaning between related terms in a language are often arbitrary and particular to that language. Sapir and Whorf took this one step further by arguing that a person's world view is largely determined by the vocabulary and syntax available in his or her language.
The extreme ("Weltanschauung") version of this idea, that all thought is constrained by language, can be disproved through personal experience: all people have occasional difficulty expressing themselves due to constraints in the language, and are conscious that the language isn't adequate for what they mean. Perhaps they say or write something, and then think "that's not quite what I meant to say" or perhaps they cannot find a good way to explain a concept they understand to a novice. This makes it clear that what is being thought is not a set of words, because one can understand a concept without being able to express it in words.
The opposite extreme--that language does not influence thought at all--is also widely considered to be false. For example, it has been shown that people's discrimination of similar colors can be influenced by how their language organizes color names. Another study showed that deaf children of hearing parents may fail on some cognitive tasks unrelated to hearing, while deaf children of deaf parents succeed, due to the hearing parents being less fluent in sign language. And computer programmers who know different programming languages often see the same problem in completely different ways.