If $K\subseteq L$ and $a,b\in L$, then $K(a,b)$ is by definition the smallest subfield of $L$ that contains $K$ and $a$ and $b$. On the other hand, $K(a)K(b)$ is the smallest subfield of $L$ that contains $K(a)$ and $K(b)$. These descriptions are almost identical. It is easy to prove that they are the same by proving that $K(a,b)\subseteq K(a)K(b) \subseteq K(a,b)$.
Here is a slightly more sophisticated approach. For $U\subseteq L$ any subset, $K(U)$ is the smallest subfield of $L$ that contains $K$ and $U$. Then $ K(a)K(b)=K(K(a)\cup K(b)) = K(K \cup \{a,b\}) = K(\{a,b\}) = K(a,b) $