The roots of mermaid mythology are more varied than one would expect.
The earliest known mermaid legends come from Syria where the Syrian goddess Atargatis dove into a lake to take the form of a fish, but the powers there would not allow her give up her great beauty, so only her bottom half became a fish and she kept her top half in human form.
Later tales in the mythology of mermaids stem from Homer's epic "The Odyssey", where some mythologists believe the Sirens to have been in mermaid form. This was an extremely popular version of the mermaid throughout history. Many popular tales including legends from the British Isles and the famous Arabian Nights tales identify mermaids in exactly this fashion. In these myths, mermaids would sing to men on ships or shores nearby, practically hypnotizing them with their beauty and song. Those affected would rush out to sea only to be either drowned, eaten, or otherwise sent to their doom.
The evil-intentioned mermaid is not the only way these creatures were seen as dangerous. Some believed that even well-intentioned mermaids would cause great danger to men who believed they saw a woman drowning and would dive into the waters to save them. Other tales suggest that mermaids either forgot or didn't understand that humans could not breathe underwater, and they would pull them down into the depths of the sea, accidentally drowning them in the process.