This happens at the end of Chapter 11. To be precise, Watson's heart does not leap when he reads the note. Instead, his heart leaps when he sees that there is a note, even before he has read it. We can see this from the following quote:
As I set it down again, after having examined it, my heart leaped to see that beneath it there lay a sheet of paper with writing upon it. I raised it, and this was what I read...
As you can see, his heart has leapt before he has seen the content of the note.
His heart leaps at that point because he is in the hut searching for clues. When he sees a piece of paper with writing on it, he naturally becomes excited because he thinks that he has discovered a clue that might tell him who the man in the hut is and why he has been living there.
Later, after he has read the note, Watson is excited for another reason. He now believes that he is the one who is being stalked. This of course excites him and makes him apprehensive.