Different organs were discovered at different times. A man named alexis st Martin was shot in his stomach and so he could open his stomach and his stomach acid could be removed. These were then studied and we learned about the stomach.
Another man phineas gage had a metal rod go through his head and it changed his personality and so from then on people learned about the brain.
As for today alot of research is done on mice though and so now we inject hormones and watch what happens. Or we study their organs.
Leonardo de Vinci did a lot of the early research by (illegally) cutting open cadavers and documenting what her found. This allowed him to get the shapes and locations of many of the internal organs and speculate on their functions. When the local authorities found out what he was doing, he was forced to stop (on threat of imprisonment, the church getting involved, etc etc).
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