What does St denote?
In this case it denotes arbitrary state passed from $A_0$ to $A_1$. This is a common idiom in interactive security game definitions. 

Conceptually a security definition should involve a single adversary who plays some game. The adversary is modeled as arbitrary algorithm. If there is some query/interaction adversary can do repeatedly, we often model this as giving oracle access to some function.

But suppose the game has a significant event that should happen only once (in this case, asking for and receiving a challenge ciphertext). This is not so easily modeled by providing oracle. Instead, it is common to formalize the situation like the authors have done here. $A_0$ is the part of the adversary that runs up until requesting the challenge ciphertext (it terminates by giving the two chosen plaintexts). $A_1$ is the part of the adversary that runs after receiving the ciphertext. To follow our intuition that there is just a "single adversary" this whole time, we allow $A_0$ to pass arbitrary information (without loss of generality, its entire internal state) to $A_1$. That's what $st$ is here.