"How" here refers to what qualities of the artificial chemicals cause sodas containing them not to be really good to drink.
Perhaps this is a clearer example: Initial question - "How did you know I was in London last week?" "Well, Harold saw you there, and he posted it on Facebook, and Diane saw his post and mentioned it to me yesterday at yoga." "OK, now I can see how you knew I was in London last week."
Likewise, start with the question "How are artificial chemicals in soda not really good for me?" = "How do artificial chemicals harm one's health?" After receiving an explanation: "OK, now I understand how artificial chemicals harm one's health. I can see how they are not really good for me."