Julia only looks at the triple as an end-marker — the parser isn’t keeping track of open/close “…” pairs inside the triple. Need to escape the first in the ending “”“” sequence.
"""I am a "string"""" is being parsed as """I am a "string""" followed by a ", and that is why (as at-ihnorton) said, the first " in the final sequence needs to be quoted.
If you have wish to have more than 2 " in a row in a triple-quoted string, then you need to use \ on every 3rd one, for example, to have 6 " inside a triple-quoted string, you’d need to have ""\"""\".