Let $f: X \rightarrow [0, \infty]$ be a measurable function.
For integers $n\ge 1, k \ge 0$, let $$E_{n,k} = \{x\in X: k/2^n \le f(x) \lt (k+1)/2^n\}.$$
Let $$E_{\infty} = \{x\in X: f(x) = \infty\}.$$
We denote by $\chi_{n, k}$ the characteristic function of $E_{n,k}$ and
by $\chi_{\infty}$ the characteristic function of $E_{\infty}$.
Define $$g_n = \sum_{k=1}^{\infty} \frac k{2^n} \chi_{n, k} + n\chi_{\infty}$$
and $$h_n = \sum_{k=0}^{\infty} \frac{k+1}{2^n} \chi_{n, k} + \infty\chi_{\infty}.$$
Then $0 \le g_1 \le g_2\le \cdots$ and $h_1\ge h_2\ge \cdots$
and $g_n(x) \le f(x) \le h_n(x)$ for all $n$ and $x\in X$.
By definition,
$$\int g_n\ \mathsf d\mu = \sum_{k=1}^{\infty} \frac k{2^n}\mu(E_{n,k}) + n\mu(E_{\infty}),$$
$$\int h_n\ \mathsf d\mu = \sum_{k=0}^{\infty} \frac{k+1}{2^n}\mu(E_{n,k}) + \infty\mu(E_{\infty}).$$
Lemma.
Let $f: X \rightarrow [0, \infty]$ be a measurable function.
Let $g_n, h_n$ be as defined above.
Then $$\sup \left\{\int g_n\ \mathsf d\mu : n = 1, 2, \cdots\right\} = \inf \left\{\int h_n\ \mathsf d\mu: n = 1, 2, \cdots\right\}.$$
Proof:
If $\mu(E_{\infty}) \gt 0$, then $\sup \left\{\int g_n\ \mathsf d\mu: n = 1,2, \cdots\right\} = \infty$ and $\int h_n\ \mathsf d\mu = \infty$ for all $n$. Hence we may assume $\mu(E_{\infty}) = 0$.
Suppose $\text{inf} \{\int h_n\ \mathsf d\mu: n = 1, 2, \cdots\} = \infty$.
Then $$\int h_n\ \mathsf d\mu = \sum_{k=0}^{\infty} \frac{k+1}{2^n}\mu(E_{n,k}) = \infty$$ for all $n$.
Choose an integer $n \ge 1$.
Let $$E = \{x\in X: f(x) \lt \infty\},$$
then $$E = \bigcup_{k=0}^{\infty} E_{n,k}.$$
Suppose $$\mu(E) = \sum_{k=0}^{\infty} \mu(E_{n,k}) \lt \infty.$$ Since $$\int h_n\ \ \mathsf d\mu = \sum_{k=0}^{\infty} \frac k{2^n}\mu(E_{n,k}) + \frac1{2^n} \left(\sum_{k=0}^{\infty} \mu(E_{n,k})\right) = \infty,$$ we have $$\sum_{k=0}^{\infty} \frac k{2^n}\mu(E_{n,k}) = \infty.$$
Hence $\int g_n\, d\mu = \infty$.
Thus the assertion is proved in this case.
Suppose $$\mu(E) = \sum_{k=0}^{\infty} \mu(E_{n,k}) = \infty.$$
Since $$E = \bigcup_{k=0}^{\infty} E_{n,k} = \bigcup_{m=1}^{\infty} \bigcup_{k=1}^{\infty} E_{n+m,k},$$ we have
$$\lim_{m\rightarrow \infty} \sum_{k=1}^{\infty} \mu(E_{n+m,k}) = \infty.$$
Since $$\sum_{k=1}^{\infty} \mu(E_{n+m,k})\le \int g_{n+m}\ \mathsf d\mu,$$ we have $$\lim_{m\rightarrow \infty} \int g_{n+m}\ \ \mathsf d\mu = \infty.$$
Hence $$\sup\left\{\int g_n\ \ \mathsf d\mu: n = 1, 2, \cdots\right\} = \infty.$$
Thus the assertion is also proved in this case.
Suppose $$\inf\left\{\int h_n\ \mathsf d\mu: n = 1, 2, \cdots\right\} \lt \infty.$$
There exists $n_0$ such that $$\int h_{n_0}\ \ \mathsf d\mu \lt \infty.$$
Let $n$ be any integer such that $n\ge n_0$.
Since $g_n \le h_n \le h_{n_0}$, $$\int g_n\ \mathsf d\mu \le \int h_n\ \mathsf d\mu \le \int h_{n_0}\ \mathsf d\mu \lt \infty.$$ Then $$\int h_n\ \mathsf d\mu - \int g_n\ \mathsf d\mu = \sum_{k=0}^{\infty} \frac1{2^n}\mu(E_{n,k}) = \frac1{2^n} \sum_{k=0}^{\infty} \mu(E_{n,k}).$$ Since $\sum_{k=0}^{\infty} \mu(E_{n,k}) \lt \infty$, letting $n\rightarrow \infty$ we are done. $\Box$
Now we prove your assertion $$\int^* f\ \mathsf d\mu = \sup\left\{\int s\ \mathsf d\mu: 0\le s\le f, s \text{ simple }\right\}.$$
For integers $n,m \ge 1$, define $$s_{n,m} = \sum_{k=1}^m \frac k{2^n} \chi_{n, k} + n\chi_{\infty}.$$
Then $s_{n,m}$ are simple functions and for each $n$, $\{s_{n,m}\}$ is an increasing sequence with respect to $m$. It is clear that $$\lim_{m\rightarrow \infty} \int s_{n,m}\ \mathsf d\mu = \int g_n\ \mathsf d\mu.$$
Hence by the lemma $$\sup\left\{\int s_{n,m}\ \mathsf d\mu: n, m \ge 1\right\} = \sup\{\int g_n\ \mathsf d\mu: n \ge 1\} = \inf\left\{\int h_n\ \mathsf d\mu: n \ge 1\right\}.$$
Let
\begin{align}
\alpha &= \inf\left\{\int h_n\ \mathsf d\mu: n \ge 1\right\},\\
\beta &= \sup\left\{\int s\ \mathsf d\mu: 0\le s\le f, s \text{ simple }\right\},\\
\beta' &= \sup\left\{\int s_{n,m}\ \mathsf d\mu: n, m \ge 1\right\}.
\end{align}
Then $\beta' \le \beta \le \alpha$.
Since $\beta' = \alpha$, $\beta = \alpha$.
On the other hand, $\beta \le \int^* f\ \mathsf d\mu \le \alpha$.
Hence $$\beta = \int^* f\ \mathsf d\mu.$$ $\Box$
