Note that "$\sigma$-finite" is a tricky notion. For example, any ${\rm\ II}_1$ factor, even non-separable, is $\sigma$-finite, because the finite trace is a faithful state.

In any case, the argument one needs is that of Makoto Yamashita, with a few clarifications. Note that for type I and type II factors, the assertion is trivial: for type I, the $\sigma$-finite factor can be taken to be $\mathbb{C}$. For type ${\rm\ II}_1$, it is always $\sigma$-finite. And any type ${\rm\ II}_\infty$ is the tensor of a ${\rm\ II}_1$ with a ${\rm\ I}_\infty$, so again the assertion holds.

This leaves us then with a factor of type III. The question is why does there exist a projection $p$ with $pMp$ $\sigma$-finite. It is well-known that any von Neumann algebra has a faithful semifinite normal weight. From this one can deduce that there has to exist a projection $p$ where the weight is finite. And then one deduces that, restricted to $pMp$, the weight is a faithful normal state. As $M$ is type III, one can construct a family of projections $\{p_j\}_{j\in J}$ with $p_j$ equivalent to $p$ for all $j$. This equivalences can be used to construct a system of "matrix units", from where the isomorphism
$$
M\simeq(pMp)\otimes B(\ell^2(J))
$$
follows.