Standard model doesn't predict that neutrinos are massless. It's a "Model", where initially neutrinos are considered massless, because no mass was observed.
The way we know, now, that neutrinos have masses, is through the mixing between the different neutrino types, through a matrix called the PMNS matrix (similar to the CKM matrix for quarks). This mixing explains the number of neutrinos that come from the sun, and sets a lower limit for the masses. But no one has done a real measurement for the masses of neutrinos... dealing with neutrinos is really hard.
I agree with the answer of Quantum physicist , that zero mass for neutrinos was an input to the standard model , not a prediction, because measurements showed a mass compatible with zero.
But I will add that the discovery that neutrinos must have mass does not destroy the Standard Model, just different Lagrangian for the neutrinos has to be included.
However the experimentally established phenomenon of neutrino oscillation, which mixes neutrino flavour states with neutrino mass states (analogously to CKM mixing), requires neutrinos to have nonzero masses. Massive neutrinos were originally conceived by Bruno Pontecorvo in the 1950s. Enhancing the basic framework to accommodate their mass is straightforward by adding a right-handed Lagrangian. This can be done in two ways. If, like other fundamental Standard Model particles, mass is generated by the Dirac mechanism, then the framework would require a SU(2) singlet. This particle would have no other Standard Model interactions (apart from the Yukawa interactions with the neutral component of the Higgs doublet), so is called a sterile neutrino. Or, mass can be generated by the Majorana mechanism, which would require the neutrino and antineutrino to be the same particle.
In standard model, the mass of a particle can be explain by either Dirac or Weyl equation. The first thing is that neutrinos are can't be described by any of the above equations (Dirac equation or Weyl equation) in the standard model because no right handed neutrinos are observed. Dirac equation needs four spinors to explain the mass of any particle. But in case of neutrinos, we have only two spinors. On the other hand, Weyl equations are only for the massless particles. Thus, neither the Dirac equation nor the Weyl equation can explain the neutrinos. Simply that why neutrinos are massless in standard model according to me.