I have a feeling there is a BrE vs AmE difference here. I'm sure I have seen BarbD say many times that the period always comes in the same place, but for the life of me, I can't remember whether it's inside or outside the quotation marks in AmE. I'm sure we'll find out pretty soon!
For BrE, if the entire sentence makes up the whole quote, then the full stop (period) goes inside the inverted commas. If you are only quoting part of a sentence, or perhaps just one word, then it goes outside the inverted commas.
John said "I am wearing a hat."
Several times while John was talking, he said "treading on eggshells".
I'm not sure what he meant when he said "plethora".
In the quote you posted, I would have put the full stop inside the quotation marks.