How can I construct a Karnaugh map from this table made from a Moore FSM Transition diagram?
This looks like a homework problem, and we don't just hand out homework solutions here. But I will give you some hints to get started.

You have named the states (S0, S1, etc.) the same as your state variables (S0 and S1). That is incredibly confusing, so pick different names for the states.

You need 2 Karnaugh maps, one for each of the "next state" bits. The input signals for the Karnaugh maps will be the two state variables (the "current state" bits) as well as the two input variables.

So, create your two, four-variable K maps and go through the usual procedure to find your optimized SOP implementation.