The peculiar structure with verb "piacere" you noticed is due to the fact that literal translation of this Italian verb is "to be pleasing to (someone)", so that this "someone" is the indirect object of the sentence and not the subject.
As the book Italian Verbs for Dummies explains (I have done a correction to the text because there was a confusion between the words "wine" and "milk")
As a verb, piacere (to like, to be pleasing to) is unique in that it takes indirect-object pronouns rather than personal pronouns. For example, "I like wine" becomes "Wine is pleasing to me," or "Mi piace il vino." Similarly, you need the preposition "a" (to) when using proper nouns. For example, you translate "A Domenico piace la pasta" as (literally) Pasta is pleasing to Domenick (or Domenick likes pasta).
As @DaG has pointed out in a comment, "personal pronouns" in the above extract really refers to "subject pronouns".
At this book you can also read
With the verb piacere, what's typically the object in an English construction becomes the subject in an Italian construction. For example, I like spaghetti becomes (literally) To me, spaghetti is pleasing, or "Mi piacciono gli spaghetti."
The same book also explains that an Italian verb which behaves in a similar way is "mancare", used to convey the meaning of "to miss" or "to lack":
The verb mancare, which means to miss or to lack, functions in precisely the same manner as the verb piacere (to like). In other words, you generally use it with indirect objects and indirect-object pronouns, and it translates literally as Something is missing to me.
An example of use of "mancare" which come from this book is the following:
Cara mamma e papa, mi mancate! (Dear mom and dad, I miss you!)
The literal translation of "mi mancate" would be "you (subject) are missing to me (indirect object)".
Other examples of "mancare" coming from the same book are:
Io ti manco? (Do you miss me? Literally: Am I missing to you?)
Sì, tu mi manchi. (Yes, I miss you.)
Mi manca mia madre. (I miss my mother.)