The major deduction that Watson makes when Barrymore tells him about this note is that Laura Lyons either killed Sir Charles or that she set him up to be killed.
In the note, Laura Lyons asks Sir Charles to meet her at the gate where he stood on the night that he was killed. She furthermore asks him to burn the letter after reading it. That is all of the letter that remained after he burned it. From that, Watson deduces that it is very likely that Lyons lured him out to the gate so that he could be killed.
As it turns out, Lyons' note was written for this purpose, but she did not know this. It was dictated to her by Stapleton, with whom she was in love.