It is a way of stating your preference, especially when it is in conflict with what actually happened or what is possible. It can be genuine, but sometimes it is used sarcastically, or insincerely.
Your example fits a scenario where person A has asked person B to tell them how the company can improve. Instead of answering directly, person B begins talking about all the positive things, leaving out the negative things which could be improved upon. Person A may then say:
As much as I’d like to hear great things about our company, I can assure you that I would much prefer to hear how you think we can get better.
What they mean by this is that, while they may like to hear the good things, they would prefer to hear about possible improvements. I would suggest that there is a tone of sarcasm in this statement and is rather like asking the person to answer the question the way it was asked.
Other more sincere examples might be:
As much as I would like to vacation abroad this year, I am staying in my own country.
This states that the person would like to go abroad, but for whatever reason cannot.
You could also use it in reply to a question:
-"Do you exercise?"
-"Not as much as I would like".
This is a genuine response which says that you do exercise but would like to be able to do it more frequently.