As a positivist Comte thought that all of human history was divided into stages, and that the world had, with the turmoil of the French Revolution and the rise of industrialization, entered into a "scientific" stage, which was the final stage. He believed that knowledge would continue to progress to the point where all of humanity's problems could be addressed through critical reasoning. Comte thought that it was time for the old ruling forces to step aside and be replaced by a new elite, steeped in science, to take over. His thinking was a bit like Marx, but he didn't think France's class problems would be overcome by class warfare. He rather thought the two could be bound together through mutual interest if they were rational enough to determine what those were. It's strange that you can't talk about Saint-Simon, since he was so influential in Comte's career.
You want to know about Comte then, not Saint-Simon? From what I remember, Comte was one of the founders of the field of sociology. Basically, at that time the word was changing and human interactions were getting more complex. We needed a way to explore that.