Meaning of "work better for silence"

Holmes and Watson are alone in the room.

Holmes: I shall work better for silence.

Watson: Oh, well. I dare say I can find something quiet to do.

Does he mean he needs silence to work? And why does he use for? Isn't it right to say in silence?

Yes, he's saying he would work better if it were quiet. Nowadays I think you'd see "in silence". The example is a more archaic usage, but I believe it falls under this definition of "for":

Having (the thing mentioned) as a reason or cause ... I could dance and sing for joy

Yes, it means that Holmes wants to work in silence. For is used here with the sense 'if there is'. For has many uses, of which this is just one.

In silence would mean 'without talking'; a teacher might instruct pupils to work in silence. Holmes means that he wants to work without anyone else talking. For here means 'in the condition'; I work better for knowing I am appreciated.