One possible interpretation of Connes's statement is that
up to an isomorphism, there is a unique faithful indecomposable representation of any commutative von Neumann algebra on a Hilbert space.

Indeed, the category of von Neumann algebras is contravariantly
equivalent to the category
of compact strictly localizable enhanced measurable spaces.

After extracting an enhanced measurable space $(X,M,N)$ from a commutative von Neumann algebra $A$
in this manner,
elements of $A$ can be identified with equivalence classes of bounded
measurable functions on $X$ modulo equality almost everywhere.

We can now easily describe isomorphism classes of representations
of $A$ on a Hilbert space.
Such an isomorphism class is specified by partitioning $X$
into almost disjoint (up to a negligible set) nonnegligible measurable subsets $\{X_i\}_{i∈I}$,
and assigning a distinct cardinal number $a_i$ to each element of the partition.
The corresponding Hilbert space is $$\bigoplus_{i∈I} {\rm L}^2(X_i,M_{X_i},N_{X_i})⊗{\bf C}^{a_i}$$
and $A$ acts on each summand by restricting the corresponding
bounded measurable function on $X$ to $X_i$ and then acting via multiplication
on the corresponding ${\rm L}^2$-space.
Here ${\bf C}^{a_i}$ denotes any complex Hilbert space of dimension $a_i$.

Such a representation is faithful if $a_i≥1$ for all $i$.
It is indecomposable if $a_i≤1$ for all $i$.
Thus, a faithful indecomposable representation must have $a_i=1$ for all $i$,
and there is a unique such a representation, namely ${\rm L}^2(X,M,N)$,
also known as the Haagerup standard from of $A$.

The point of all this is that although the (abstract) commutative von Neumann
algebra appears to know nothing about an enhanced measurable space
or its ${\rm L}^2$-space, all this data can be reconstructed
in a unique manner, i.e., it is unique up to a unique isomorphism.

If, furthermore, we know the Dirac operator, we can proceed to refine
$X$ to a smooth manifold, as described by Connes.