On the Wikipedia page there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to conditional expectation. Rigorously, how are the two related?
There's a nice discussion of these issues in "Conditionng as Disintegration" by Chang & Pollard: https://onlinelibrary.wiley.com/doi/full/10.1111/1467-9574.00056
The example one should have in mind is the Fubini theorem for the unit square (endowed with the Lebesgue measure) projected onto the horizontal base. The conditional measures are then just the Lebesgue measures on the vertical intervals. According to the Rokhlin theory, this is actually the general case, namely, measure preserving maps between reasonable probability spaces are precisely like this (up to an isomorphism). Have a look at the section "Regular conditional probabilities" of the Standard probability space entry (it is written unusually well for Wikipedia) and the references therein.