  We report on the discovery and localization of fast radio bursts (FRBs) from
the MeerTRAP project, a commensal fast radio transient-detection programme at
MeerKAT in South Africa. Our hybrid approach combines a coherent search with an
average field-of-view of 0.4 $\rm deg^{2}$ with an incoherent search utilizing
a field-of-view of $\sim$1.27 $\rm deg^{2}$ (both at 1284~MHz). Here, we
present results on the first three FRBs: FRB 20200413A (DM=1990.05 pc
cm$^{-3}$), FRB 20200915A (DM=740.65 pc cm$^{-3}$), and FRB 20201123A
(DM=433.55 pc cm$^{-3}$). FRB 20200413A was discovered only in the incoherent
beam. FRB 20200915A (also discovered only in the incoherent beam) shows
speckled emission in the dynamic spectrum which cannot be explained by
interstellar scintillation in our Galaxy or plasma lensing, and might be
intrinsic to the source. FRB 20201123A shows a faint post-cursor burst about
200 ms after the main burst and warrants further follow-up to confirm whether
it is a repeating FRB. FRB 20201123A also exhibits significant temporal
broadening consistent with scattering by a turbulent medium. The broadening
exceeds that predicted for medium along the sightline through our Galaxy. We
associate this scattering with the turbulent medium in the environment of the
FRB in the host galaxy. Within the approximately $1'$ localization region of
FRB 20201123A, we identify one luminous galaxy ($r \approx 15.67$;
J173438.35$-$504550.4) that dominates the posterior probability for a host
association. The galaxy's measured properties are consistent with other FRB
hosts with secure associations.
