The following is an excerpt from A Little History of Science.
If you want to really understand how something is made, it is often a good idea to take it apart, piece by piece. With some things, like watches and cars, it helps if you also know how to put them back together again. If what you want to understand is a human or an animal body, it has to be dead before you start, but the goal is the same.

Galen, as we know, dissected – took apart – many animals, because he couldn’t dissect any humans. He assumed that the anatomy of pigs or monkeys was pretty much like that of human beings, and in some ways he was right, but there are differences, too. The dissection of human bodies started to be done occasionally around 1300, when medical schools began to teach anatomy. At first, when people noticed any differences between what they saw in the human body and what Galen had said, they assumed that human beings had simply changed, not that Galen had been wrong! But as they began to look more closely, anatomists discovered more and more small differences. It became obvious that there was more to uncover about the human body.
The man who did the uncovering was an anatomist and surgeon known to us as Andreas Vesalius (1514–64). His full name was Andreas Wytinck van Wesel. He was born in Brussels, in modern-day Belgium, where his father was a medical man employed by the German Emperor Charles V. A clever child, he was sent to the University of Louvain to study arts subjects, but decided to change to medicine. Clearly ambitious, he then went to Paris where some of the best teachers were. They all followed Galen, and during his three years there he impressed them. He also showed his abilities in Greek and Latin, and his fascination with dissection. A war between the German Empire and France forced him to leave Paris, but he reintroduced human dissection to the medical faculty at Louvain before travelling, in 1537, to what at the time was the best medical school in the world, at the University of Padua in Italy. He took his exams, passed with the highest distinction, and the next day was appointed as a lecturer in surgery and anatomy. At Padua they knew when they were on to a good thing: Vesalius taught anatomy through his own dissections, the students loved him, and the very next year he published a series of beautiful anatomical illustrations of parts of the human body. They were so good that doctors all over Europe began copying these pictures for their own use, much to Vesalius’s annoyance, since they were actually stealing his work.
Andreas Vesalius' De Humani Corporis Fabrica, p. 559
Cutting open a dead body is not a particularly pleasant thing to do. After death, the body quickly begins to decay and smell and, in Vesalius’s time, there was no way to stop it from rotting. This meant that the dissection had to be done quickly, and in an order that made it possible to get it done before the smells became overpowering. The belly was done first, since the intestines are the first to rot. This was followed by the head and brain, then the heart, lungs, and other organs in the chest cavity. The arms and legs were saved to the end: they lasted the best. The whole thing had to be done in two or three days, and anatomy was generally taught in winter, when the colder weather at least delayed the decay and gave the doctors a little more time.