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.