This is the answer I gave to a similar question asked some years ago about how the ancient Greeks and Romans interpreted fossils. Hopefully those more knowledgeable about later eras will chime in about how their folks viewed the things.

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The ancients found fossils with some regularity. Clearly aquatic fossils were taken as evidence for past floods or long-lost seas. Herodotus, for example, uses seashells found in the Egyptian desert as evidence for his theory that Egypt was created by silt deposited from the Nile (2.12.1). How else could they have gotten there if the land that is now Egypt was not once sea? Like many other cultures, the Greeks and Romans possessed a myth of a cataclysmic flood, and this presumably helped explain shells and the like outside of Egypt.

Large fossils were occasionally thought to have come from sea-monsters. At least one Roman emperor exhibited the skeleton of a sea-monster to the Roman public. This might have simply been a whale skeleton, or it may have been a fossilized creature of some sort. Connections of fossils to dragons are somewhat thin on the ground. In Philostratus' Life of Apollonius of Tyana, in the sage Apollonius spends some time in a city of dragon hunters who kill sleeping dragons to get the gemstones inside their skulls. Sometimes fossilized skulls will have crystalline formations inside of them, created by minerals leaching out of the bone. This may be the origin of the gemstones in dragon skulls idea. But since Philostratus thinks the dragons are very much alive, the connection between monster and fossil is somewhat tenuous.

Much more common are instances where the ancients interpreted large bones as the remains of dead heroes or giants. In Herodotus, the Spartans discover the giant body of the hero Orestes beneath a blacksmith's shop in Tegea and rebury it with honors in their homeland (Hdt. 1.68). We learn of a number of such giant skeletons Philostratus' Heroikos:

But do not yet regard as credible what I have said, my guest, until you sail to the island of Cos, where the bones of earthborn men lie, the first descendants of Merops, they say, and until you see the bones of Hyllos, son of Herakles, in Phrygia and, by Zeus, those of the Alôadai in Thessaly, since they are really nine fathoms long and exactly as they are celebrated in song. The Neapolitans living in Italy consider the bones of Alkyoneus a marvel. They say that many giants were thrown down there, and Mount Vesuvius smolders over them. Indeed in Pallênê, which the poets call “Phlegra,” the earth holds many such bodies of giants encamped there, and rainstorms and earthquakes uncover many others. Not even a shepherd ventures at midday to that place of clattering phantoms which rage there. Disbelief in such things probably existed even at the time of Herakles. Hence, after he killed Geryon in Erytheia and was alleged to have encountered the most enormous creature, Herakles dedicated its bones at Olympia so that his contest would not be disbelieved.

Philostratus goes on to list other examples. The important thing to note is that Herodotus' Orestes is not a one-off: people around the Mediterranean thought they found larger than life human remains, and some of these could be visited in various temples and shrines. So it may be best to take such stories as having a core of truth. People really did find large bones, and usually took them to be human.

There may be a few reasons why fossils were more commonly seen as humans rather than monsters or dragons. First, a good portion of the bones found were of Ice Age mammals, various types of pachyderms, rhinos, etc. Second, bones were usually found in a jumbled state. To quote Philostratus' Heroikos again:

But the corpse that came to light on Lemnos, which Menekratês of Steiria found, was very big, and I saw it a year ago when I sailed from Imbros, only a short distance from Lemnos, however, no longer appear in their proper order: the vertebrae lie separated from each other, tossed about by earthquakes, I suppose, and the ribs are wrenched out of the vertebrae. But if one imagines the bones together as a whole, the size seems to make one shudder and is not easily described.

So if the skeleton is dis-articulated and incomplete, and if its of a large mammal with many individual bones (molars, thigh bones, vertebrae, ribs) that bear a passing resemblance to human bones, then it is little wonder that ancients by and large took the bones to be human.