You actually think in pictures, not words. The mind is highly visual.
When you're awake and recalling a dream, you're subconsciously encoding pictures into words that your conscious mind can understand. Even when you speak to someone in your dream, you're only seeing a picture (or video, if you will) of yourself speaking to someone. Your mind fills in the concepts and ideas of the content of the conversation, but you don't actually "hear" the words, per se.
One of my Japanese friends can't speak English, so when I speak to him, we converse in Japanese. I have another friend who's American and can understand some Japanese, but can't speak it very well. So when we all get together, our conversations are a mix between Japanese and English. One time we were planning a road trip. Later on that evening after my Japanese friend had gone home, the American friend said "oh before I forget, we should stop by
Hakone and check out the hot springs sometime during the trip." And I said "Don't you remember us having a whole conversation about that? We're already planning to go there!" He insisted that I never mentioned it, but I swore that I had talked about it and our Japanese friend was also excited about going to Hakone. Then he said "Oh, then maybe you were talking about it in Japanese and I didn't understand." When he said that, I thought about it, and I realized that I couldn't remember if I had that conversation in Japanese or English. I distinctly remembered
having the conversation, but I just couldn't recall hearing the words in either language in particular. It actually happens quite often, and I'm sure that others who are bilingual or multilingual have the same experience.
If dreaming required language, it would preclude dreaming in other mammals. But other mammals dream as well. My dog dreams all the time. He twitches, whines and barks in his sleep. What language do you think animals dream in? The answer is, they don't dream in any language. And neither do we.
I actually first learned this back when I took intro to psychology in university.