These days you very occasionally hear of a camper getting attacked and/or killed by a bear, a trail jogger or mountain biker getting attacked/killed by a cougar in a state park, etc. But we rarely venture into the deep of what remains of their habitat and the attacks are very rare even when we do.
When I think of early colonists coming over from, say, Britain, these would have been people unused to large predatory animals for the most part, because they didn't have many. Bears, mountain lions, etc. I think there were still a small number of wolves on Great Britain at that time, but I believe most had been hunted down for bounties and I don't think they are normally attackers anyway. And bears were extinct over there and big cats apparently only the subject of occasional sasquatch-style rumor.
But regardless, you've got people coming over and making settlements in rural and forested areas right smack in the habitats of these big animals. I'm thinking there would have been a lot of bears and big cats and whatnot, depending on the terrain. And the people would have been out in those woods hunting game, and even if not would have been in very close proximity to the surrounding wild.
Is there much record of these early settlers being attacked and/or eaten by these kinds of animals back in colonial days?
Same question for later people out on the frontier as America pushed westward.
Governor Bradford only makes a single mention of wolves in the context of actual encounters. While scouting for a location to set up their colony some were heard howling in the darkness and frightened off with gunfire.
So being very weary, they betooke them to rest. But aboute midnight, they heard a hideous and great crIe, and their sentinell caled, "Arme, arme"; so they bestired them and stood to their armes, and shote of a cupple of moskets, and then the noys seased. They concluded it was a companie of wolves, or such like willd beasts; for ove of the sea men tould them he had often heard shuch a noyse in New-found land. 127
http://mith.umd.edu//eada/html/display.php?docs=bradford_history.xml
Governor Winthrop, leader of the later Puritan settlements in Massachusetts mentions wolves in 1630 and 1631 respectively.
1630
The wolves killed six calves at Salem, and they killed one wolf.
1631
The governor, being at his farm house at Mistick, walked out after supper, and took a piece in his hand suppos- ing he might see a wolf, (for they came daily about the house, and killed swine and calves, etc. ;) and, being about half a mile off, it grew suddenly dark, so as, in coming home, he mistook his path, and went till he came to a Uttle house of Sagamore John, which stood empty.
https://archive.org/stream/winthropsjournal00wint/winthropsjournal00wint_djvu.txt
Bears go unmentioned in either account. They are more likely to be mentioned in the accounts of settlers in the Carolinas, bears are common in the swamps of North Carolina/Southern Virginia.
The only reference to Lions is a ship by that name trading with the settlers.