There are various ways of emphasizing sppech and writing in English. In some cases, you may insert an auxiliary verb where one would not normally be necessary:
I do like your shoes.
I do appreciate.
I did phone him!
Hope that help you!
As a native speaker, I would say that "I do appreciate" normally carries a sense of contradictory emphasis. For example:
"I appreciate the the gift you've given me."
"I do appreciate the the thought, but I already I have one of these."
In the latter instance, the "do" serves to emphasise the main verb "appreciate", but the emphasis is added as a counter to the negative expression that follows. In my experience, this is the sort of context in which "do appreciate" is most often used.