It is a (bit?) of an archaic use, you can think of it as a short form of
Now kindly see (to the needs of) my man.
The fact that your example uses "good sir" and "my man" (meaning "my man servant") are an indication that this line was not written recently, or that at least the story is playing in anything but recent years - it is quite archaic.
The meaning of "see" in this way is now - I think - only used in reference to doctors. (And even then it might be confused with "to see" as "to meet face to face"):
I asked the doctor when he could see me.
After lunch, the doctor announced he would see his next patient.
If you would nowadays tell me
I will not be at home this afternoon, please see my friends kindly.
I would interpret it as:
You may want to visit me, but I will not be at home this afternoon, please understand that and instead, go visit my friends.
If you are implying I should see to your friends needs and you would express that in such a way, I might actually be offended as it is a turn of phrase that would have been aimed at a servant. In your original example, even though the person spoken to is addressed as "good sir", the request to see my man servant cannot be addressed to anyone but another servant since I am instructing him to take care of my servant.