I have come across such sentences many times. And i do the last words' meaning too , but i don't get "to my's" meaning.
I have had this problem too. It is kind of a weird construction. If you say,
She had gained five pounds over the winter, much to her chagrin/dismay/etc.
Because she had gained five pounds over the winter, she experienced chagrin/dismay/discomfort.
To my surprise, he didn't come to my birthday party.
Because he didn't come to my birthday party, I was surprised.
I'm not astonished that sentence introductions such as
to my surprise/astonishment/delight/horror etc
may sound a bit foreign to some. Actually this is a Latin formula, called dativus finalis, describing the effect. Even students who learn Latin have to learn such formulas as something special.
Such literary sentence introductions are not infrequent, but a good term for these formulas is lacking. The Longman Dictionary DCE has these expressions in "to", no. 16 and explains "someone's reaction/or feeling to an event". So one might invent the term "dative of reaction".