What Murray-von Neumann did was to show that there is an infinite-dimensional generalization of the following fact.
If $\mathcal H$ is finite-dimensional and $\mathcal M\subset\mathcal B(\mathcal H)$ is a von Neumann algebra, it is a basic exercise that we can see $\mathcal B(\mathcal H)$ as $M_n(\mathbb C)$ for $n=\dim\mathcal H$. And in that situation, $\mathcal M$ is isomorphic to $$\tag{*}\bigoplus_{k=1}^{\ell} M_{n(k)}(\mathbb C),$$ where the blocks are given by the minimal central projections (i.e., each block is $P\mathcal M P$, with $P$ a minimal central projection.
When $\mathcal H$ is infinite-dimensional, the same idea works. Thing is, now the centre may not have minimal projections, but what they proved is that there exists a Borel space $X$ and a Borel measure $\mu$ such that
$$
\mathcal M=\int_X^{\oplus} \mathcal M_\lambda\,d\mu(\lambda),
$$
where the function $\lambda\longmapsto \mathcal M_\lambda$ is factor-valued a.e.
In the particular case when $X$ is finite, $\mu$ is the counting measure and the $\mathcal M_\lambda$ are finite-dimensional, one recovers $(*)$.