Why is this greedy algorithm claim true?
The claim is not that

$$f_S(B) \ge f_S(C).$$

Rather, the claim is that

$$f_{S_1}(B) \ge f_{S_1}(C).$$

Do you see the difference?  Here $S_i$ is the set of elements chosen (by the greedy algorithm) in the first $i$ iterations, so $S_i=\{A\}$.  Thus, the claim is that

$$f_{\{A\}}(B) \ge f_{\{A\}}(C),$$

i.e.,

$$f(\{A\} \cup \{B\}) \ge f(\{A\} \cup \{C\}).$$

The latter claim is indeed true, as both the left-hand side and the right-hand side are 9 (i.e., $f(\{A,B\})=f(\{A,C\})=9$), since they cover 9 of the "blue dots".