This answer I gave a while back seems relevant here.

In short, pretty much all civilizations have tried to record their past in way or another. In some cases this even came really close to what we call historiography today. However, what we call history, the systematic study of the past and the mechanisms that move it, is a much more modern invention that originated in the West around the 18th and 19th centuries.

The word history comes from the Greek word ἱστορία and is the etymology for the word story and one definition is just the study of all things in the past. Focusing just on humans here, when we think of the word Prehistoric (meaning before/pre history) we often think of the Stone Age based on the early tools/technology used. The Stone Age can be broken down into Paleolithic, Mesolithic and Neolithic. This kind of study is considered pre history, again meaning pre a written history, things that happened before the invention of writing. So, with this second definition, scholars and historians study of history really begins with the first written records. We start tracking history starting around 3400 BCE, with the first early writings from Sumer, located in southern Mesopotamia, (in the area of modern day Iraq) which is believed to be where written language was first invented.

Writing began, as most are familiar, with picture words called pictographs and they were drawn on a clay tablet with some kind of pointed tool (later a stylus), and writing was first developed as a way to keep track of food rations. Scholars now believe that writing may have developed independently in four different civilizations. Mesopotamia first, then Egypt shortly after, China around 1200 BCE and lastly the Mesoamericas. I just wanted to add this and realize you are probably asking when people started studying history and not when we consider history to have started, unfortunately that is not my wheelhouse and luckily someone knowledgeable answered that. Edit: Thomson, James. An Introduction to the History of History.