The claim isn't that it's impossible to have a thought unless you have language for it, but rather that having language for something makes it easier. It's a close relative of Kolmogorov Complexity and a variety of theorems from machine learning regarding hypothesis classes - different languages have more or less effective ways to express the same concept, and choosing a language with better notation for a given topic can make that topic easier to handle. In many cases (especially technical topics) this change makes the difference between the topic being completely intractable and being a two-week unit in eighth grade.

As a concrete example, consider mathematics. I can mostly handle the theory behind integration by eyeballing it and using spatial intuition, but I can't get any farther than that without introducing language that allows me to precisely circumscribe concepts and relate them precisely.

edit: also note that "the thoughts that I'm having don't use words" doesn't mean that those thoughts aren't using language. It's just not a language that you could serialize to something that could come out of your mouth - it's some private, unique internal representation that your brain has constructed for it to work with concepts on its own. Still a language, for the purposes of this discussion, since that internal representation will have all the same properties as above - some people have better internal representations for particular topics.