Quotation marks are used to mention the exact words spoken during a conversation.
So basically it can be understood as the written form of verbal communication.
To have a "place at university" means that a university is willing to accept you as a student.
Now it is very natural to understand when that conversation was happening the other person had not yet joined the university.
And to talk about things that haven't happened yet, we use the future tense.
Specifically we use the simple future tense to talk about things that will start and end in future.
Looking at the alternatives:
--has studied : it is grammatically inappropriate here; because "has" is used with 3rd person not first person; "studied" indicates past and not future.
--am
going to study : future tense; "going to" indicates the intention / plan
of the speaker; and with that we use the base form of the verb. Please note that we use "going to" when talking about a decision that was made prior to the conversation.
--study : base / root form of the verb
--will study : "will" is used in sentences of future tense, and with it we use the base / root form of the verb. Now here it is important to know that we use "will" when we are making a decision or a prediction.
"That's right" in the second sentence suggests us that the other person had already made the decision of studying at St. Andrews.
Thus, the grammatically correct alternative is "am going to study".
Therefore the correct complete sentence is:
"Jane told me you have a place at university."
"That's right. I AM GOING TO STUDY Math at St. Andrews in Scotland."