What education does/did you have to go through to become a Anesthesiologist?

I did exactly as you described, with the last 4 yrs being residency, when you actually learn how to practice your chosen specialty.

Licensure is different from training; it's a state thing and involves proving that you completed med school and passed the USMLE.

Board certification in anesthesiology requires written and oral exams, and is done after completing residency.

Yep, 4 years undergraduate (usually), 4 years of medical school and then you graduate and are officially a physician. However you aren't certified to practice medicine in anything because you haven't chosen and completed the training required to specialize. That specialization is called 'residency' and it takes 4-5 years. It's on the job training with older more experienced physicians. If you wanted to become an anesthesiologist, then you would have to complete this residency in anesthesiology to get board certified as an anesthesiologist.

It's a long and difficult path and I would strongly suggest making sure it's what you want before embarking on it.