The Standard Model doesn't 'say' anything. It is a model, that we built, so we have to tell it something first for it to know about said thing.
There was no need for the neutrions to be massive when the model was built, so no mass terms were put into it's Lagrangian.
Then, some years later, these neutrino osciallation experiments were recorded.
As a side note, the experiments were actually looking for proton decays, but ended up finding none of them, and instead finding neutrino oscillations.
So the model builders went back to the drawing board, did some maths, and came up with a description of the oscillations. They found that the probability to measure a neutrino oscillation goes as
$$ P \propto \sin^2\left( \Delta m \right) $$
where $\Delta m$ is the difference in the mass of the two neutrinos (the one that it 'used to be' and the one that it oscillates to).
So we see that in order for this probability to be non-zero, we must have $\Delta m$ is non-zero.
But in order for $\Delta m$ to be non-zero the neutrino masses have to be non-zero to begin with!
The difference between zero and zero is certainly not non-zero!
So this is why they concluded that, in fact, the neutrinos must be massive, albeit it very slim.
Then they 'told' this information to the Standard Model, and the Standard Model now has non-massless neutrinos.
So in brief, the probability for an oscillation depends on the (difference in the) mass(es) of the neutrinos, which is what lets us conclude that they are not massless. Furthermore the Standard Model cannot 'explain' anything. It only repeats what we tell it (though one must admit that it does a fairly good job of it too!).