Why does Bob tremble when he finishes reading the note?

We aren't told specifically what it is that makes Bob's hand tremble when he reads the note from his old friend. However, we can certainly intuit that he is feeling a lot of different emotions in this moment, all of them strong and unexpected. Having traveled a thousand miles to meet an old friend, he has bumped into a policeman. The policeman then leaves him; shortly afterwards, another man turns up, claiming to be the old friend, Jimmy. Only when the pair emerge into the light does Bob realize that his companion cannot be Jimmy, whereupon the imposter reveals that he is actually a policeman, and hands Bob the note.
The note says that the first policeman was in fact Bob's friend Jimmy, who recognized Bob at once as a man wanted in Chicago for numerous crimes. Unable to arrest him himself, he went and fetched a colleague to do it instead. So, Bob's hand may be trembling because of shock or perhaps because of a sense of betrayal that his old friend would do something like this to him. There may also be an element of anger at Jimmy and fear of what is to come now that he has finally been apprehended by the police.

In "After Twenty Years," two friends have dinner together. They decide to meet in that exact location in exactly twenty years. The men go their separate ways to seek their fortunes and make lives for themselves. Bob travels west and his friend, Jimmy, remains in New York. After twenty years have gone by, Bob waits in the agreed upon location. A cop passes by, and Bob tells him the story of how he is there to see his friend from long ago. The cop listens to the story and then continues along the street.
A short time later, a man claiming to be Jimmy appears. After they walk together a short distance, Bob discovers that the man is not his friend. Explaining that he is a cop, the man hands Bob a letter from Jimmy. As he reads the letter, Bob's hand begins to shake. The author does not explicitly state why Bob's hand shakes. However, one can speculate that perhaps his hand shakes because he learns that the cop he first speaks with is his friend from twenty years ago. Bob realizes that his success is about to come to an end because he will be arrested. His friend does not have the heart to arrest him but will see to it that Bob is held accountable.