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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.