Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable Let's consider the space $\mathcal{L}^p(X, \Sigma, \mu)$ of all functions $f\colon X \to \mathbb{R}$ (or $\mathbb{C}$) for which:
$$
\int\limits_X|f|^p \mu(dx) &lt; \infty.
$$
Here $X$ is a metric space, $\Sigma\subset 2^X$ — sigma-algebra on $X$ and $\mu$ — is the measure on $\Sigma$. As usual, functions that are equal almost everywhere considered to be equivalent, so technically we are dealing with classes $$[f]=\{g\in \mathcal{L}^\infty(X,\Sigma, \mu) \colon \mu \{x\in X\colon f(x)\neq g(x)\}=0\}.$$ This space is endowed with integral metric $$\rho_p(f,g)=\left(\int\limits_X|f-g|^p \mu(dx)\right)^{1/p}$$

The statement is that space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable, where $\forall A,B\in\Sigma$ $$\rho_\Delta(A,B)=\mu(A\Delta B)$$
Here, again, we consider $A\in \Sigma$ as a class $[A]=\{A'\in\Sigma\colon \mu(A\Delta A')=0\}$ to have the identity of indiscernibles.

How do one prove the statement?
As it has been pointed out by Ramiro, the assertion to be proved in the question is false. Only one of the implications is correct that is if $(\Sigma, \rho_{\Delta})$ is separable, then $L^p(X,\Sigma, \mu)$ is separable. This is what will be shown below.

Let $\chi_A$ be the characteristic function of the set $A$. Now observe that 

$\| \chi_A - \chi_B\|_{L^p}^p = \int|\chi_A - \chi_B|^p = \int_{A\setminus B}1 + \int_{B\setminus A}1 = \mu(A\setminus B) + \mu(B\setminus A) = \rho_{\triangle}(A,B)$

Now consider the function $\varphi: \Sigma \to L^p(X, \Sigma,\mu )$ defined as $ A \mapsto \chi_A$.

This is an injective map from $(\Sigma, \rho_{\triangle})$ to $(L^p(X, \Sigma,\mu), \rho_p)$.

Observe that $\rho_{\triangle}(A_i,B) \rightarrow 0 \Leftrightarrow \|\chi_{A_i} - \chi_{B} \|_{L^p} \rightarrow 0$. 

Consider a countable dense subset in $L^p$ say $f_i$. Now choose an element from each of the following sets $B(f_i,1/k)\cap \mathrm{Im}(\varphi)$ say $\chi_{A_{i,k}}$ where $i$ and $k$ vary over natural numbers. This will be a countable subset of $\mathrm{Im}(\varphi)$. For any given $\varepsilon$ and $\chi_A$ there exists $f_j$ such that $\| f_j - \chi_A \|_{L^p} &lt; 1/k &lt; \varepsilon/2$. Now let's choose $\chi_{A_{j,2k}}$, then $\| \chi_{A_{j,2k}} - \chi_A \|_{L^p} \leq \varepsilon $. Mistake in this argument was also pointed out by Glinka below.

Let $\lbrace A_i \rbrace$ to be a dense subset in $\Sigma$. Now consider the vector space spanned by $\mathbb{Q}\chi_{A_i}$. This is a countable set. This is dense in the vector space spanned by $\mathbb{R}\chi_{A_i}$ which is dense in $L^p$ since simple functions are dense in $L^p$. One can suitably modify the proof for complex valued functions.
The correct statement is that the space of summable functions (modulo a.e. equivalence) is separable if and only if there is a countable collection C of measurable sets of finite measure such that for each measurable set Q of FINITE measure and each integer N there is a set A in the collection C such that the measure of the symmetric difference between Q and A is less that 1/N. I have been unable to find a published proof of this statement, but the assertion is made in a book by Zaanen on measure theory published around 1967. However, the proof of the tricky direction made there is wrong. I know how to prove this result. If readers are interested I will provide a proof here.