Epsilon Perfect Secrecy: Size of Key Space
Choose a uniform distribution over the message space $M$. Suppose we observe a ciphertext $c$. Let $M_c = \{ m | \exists k. dec_k(c) = m \}$ be a set of all messages that could have been encrypted to $c$.
Choose a message $m \in M_c$ for which $\Pr[m|c] \geq \frac{1}{|K|}$ (such message exists, because the set $M_c$ has at most $|K|$ elements). Without knowing $c$, the probability of $m$ being chosen is $\Pr[m] = \frac{1}{|M|}$. Thus, we conclude that $\frac{1}{|K|} \leq \Pr[m|c] \leq (1+\epsilon)\Pr[m] = (1+\epsilon)\frac{1}{|M|}$, from which your desired inequality follows.