To hypothesize means simply to make a hypothesis. Which is just a scientific way of saying "make a really good educated guess."
Ok, so when someone hypothesizes there's a little bit more involved than just guesswork. It involves using your past knowledge and available facts to try and predict what might happen. Scientists hypothesize all the time — it's the basis for every experiment. For example, doctors examining a werewolf might hypothesize that his furry face and oddball behavior stem from an illness and not the full moon.