Their specific reasons, from the linked paper, include:
Use of the random-effects approach in modelling co-twin control data
is intuitively appealing. We can think of the responses from each
member of an MZ twin pair as repeated measures for one pair. The
random-effects model assumes that the logit varies from one pair to
the next by $\nu_i$. This assumption is reasonable because each pair of
twins has its own unique shared genetic and environmental background.
Thus, this variability reflects natural heterogeneity due to
unmeasured genetic and environmental factors among all the twin pairs.
This heterogeneity is represented by a Gausssian probability
distribution, that is, the random pair effect, $\nu_i$, which is assumed
normally distributed in the population.
One important feature of the random-effects approach is that it can
readily accommodate missing data in the response variables.
In random-effects models, the effects of the individual-level
confounding variables also involve both within-pair and between-pair
comparisons. Thus, they may differ from the effects estimated through
the conditional likelihood approach. In addition, we can estimate the
effects of pair-level variables (for example, age) with use of the
random-effects approach, but not with use of the conditional
likelihood approach, though we can estimate the interactions between
pair-level variables and the exposure using both approaches.
Do you have particular disagreements with these or are they unclear? Or are you looking for general principles that are not particularly tied to this study at all?
In terms of generally why you'd use a random effect, that's pretty complex and unfortunately different fields define "fixed effect" and "random effect" differently. There are also different names given to mixed-effects modeling, including "hierarchical" modeling. (Not to mention a lot of mixed-effects modeling now-a-days is Bayesian rather than Frequentist.)
In this particular study, they were looking at intra- and inter-twin-pair differences, so mixed-effects models naturally come to mind. As I understand it, a fixed-effect model will either pool completely across all observations (if you leave out a group variable), or will not pool at all across observations (if you include a group variable). A mixed-effects model allows for limited pooling, driven by the data. ("Borrowing strength" is the term.) Not sure how much borrowing strength applies in a paired study like this, though.
Last, since they are treating twins as repeated measurements, their errors will be correlated, which violates the independence assumption of a fixed-effects model.