Why is this? What are the advantaged of imposing a Majorana condition in this context?
In this context, the word "Majorana" doesn't mean that it is the "real one-half" of the ordinary simple Dirac spinor. It means that it is any half-integer field with a reality (Majorana) condition!
For example, in the Chapter 4 of the DAMTP lectures, the field introduced is a standard spin-3/2 field with one spinor index and one vector index, as required for a gravitino. The gravitino's spin has to be 3/2 because it differs by 1/2 from $j=2$ of the gravitons, and $j=5/2$ is already too much and would require too large gauge invariance.
The Majorana reality condition is imposed on the gravitinos simply because the metric tensor is naturally a real (or, quantum mechanically, Hermitian) field which is why it must be possible to naturally impose the reality condition on the superpartner field, the gravitino field, too.
The minimum $N=1$ supersymmetry is generated by 4 real supercharges which may be imagined to combine to one Weyl (complex chiral) spinor or one Majorana (real non-chiral) spinor. Using the latter approach, the gravitino superpartners of the real graviton by the real supercharges are real, too.
Analogously, we usually describe the gauginos, superpartners of (naturally real) gauge bosons, as Majorana fermions while the superpartners of (complex but not only complex) scalar fields are naturally Weyl fermions – even though the information in one Weyl fermion and one Majorana fermion is really "the same".