The conscious mind is the part of your mind that is in the forefront - what you're actively thinking about, what you're aware of. Your unconscious mind is more the part of you that is behind the scenes, that can influence how you feel/think/etc. but isn't active in your mind at one time.
In this case, it means he can cause Fischer to think (actively) concepts that he wants him to think, not just affect his mood or something like that which would be in the 'unconscious' mind.
Freud has a concept of this; explained in this article:
In Sigmund Freud's psychoanalytic theory of personality, the conscious mind consists of everything inside of our awareness. This is the aspect of our mental processing that we can think and talk about in a rational way.
versus the unconscious mind:
Things that the conscious mind wants to keep hidden from awareness are repressed into the unconscious mind. While we are unaware of these feelings, thoughts, urges and emotions, Freud believed that the unconscious mind could still have an influence on our behavior.
He emphasized conscious here because it's more significant that he can suggest things in the conscious mind than in the unconscious mind. Many people believe that you can influence others' unconscious mind through various techniques (change the music in the room, change the lighting, who knows), which then has a later but more subtle effect on the conscious mind. Directly suggesting things to the conscious mind is basically telepathy, which is a significant thing and can be very powerful, especially if it could be done without the subjects' awareness of the change.