I believe the phrase "of note" in this case is an appositive adverbial phrase modifying the main clause ("these associations were generally quite strong"). It attempts to give you a better description or context for the main conclusions.
Your instinct to replace it with "importantly" is also correct, and "it is of note that" is even more literal, however it is not uncommon or incorrect for speakers/writers to shorten these phrases to the barest minimum of words that retain meaning. It's just a matter of style.
Consider the following:
"Dinner finished, we left the house."
"Having finished dinner, we left the house."
"After we finished dinner, we left the house"
In all three, the first phrase (about dinner) is a modifier of the second (it describes the time/circumstances of when we left the house). The first conveys this idea in the fewest possible words, followed by the second, and then the third. All are grammatically correct and mean the same thing, it's just a stylistic choice by the author of which to use.
In the example you give, "of note..." is the same idea. The full literal sense of the phrase is, "it is of note that..." but it has been shortened down to a more compact form.
There is nothing grammatically incorrect about starting a sentence in this way.