I assume that many ancient civilizations built their settlements and cities right on top of previous ones for the sake of convenience. But presumably at some point, someone said "Hey: maybe we should preserve and investigate what came before". When and why did that occur?
If you are talking about archaeology as the rudimentary study of "things dug up" for the purposes of developing an historical narrative about a people, then I think most would agree that there was archeology going on by the early to mid-nineteenth century. Archaeology has undergone a number of changes in methodological and theoretical orientation. And one might argue that the methods were so lacking and a theoretical orientation so absent in early studies that they were not in the same discipline we appreciate today. But the consensus seems to be that archeological investigations that collected and reported data on reconstructing cultural activity widely appear in the mid 1800s.
This figure (fig. 1, pg. 11, Important Movements in Archaeology) of Triggers study shows the timing and emergence of many types of archaeological inquiry. Edit: Note that I did not include antiquarianism in this discussion because I do not believe it is sufficiently related to archaeology.
Willey and Sabloff, History of American Archaeology
Bruce Trigger, History of Archaeological Thought
There is a short bit in one of my archaeology textbooks about this.
Around 4000 years ago the ancient Egyptians excavated and reconstructed the sphinx which had been build 500 years earlier.
Around 2500 years ago the king of Babylon excavated the remains of a ruined building to determine its age and its relationship to Babylon. He also displayed some of the things recovered in his palace.
The Romans would also collect ancient art. Especially ancient Greek art. They too a great interest in the ancient Greeks. They would also travel to ancient ruins as tourists.
Heinrich Schliemann had claimed to have discovered the site of the Trojan War in the 1870's, and had claimed to have found "Priam's Treasure". He made claims and used methods that are highly dubious by today's standards, although this had helped popularize archeology at the time (and thereby becoming a thing).
"Archaeology and the Iliad: The Trojan War in Homer and History" by Eric H. Cline, from the Modern Scholar series, covers this in some depth.