By themselves, they don't. But quantum probabilities are unique. They cannot be explained by missing information about local, pre-existing properties. Therefore, to maintain determinism, you are forced to accept extreme alternatives like non-locality (actions here instantly affecting the other side of the universe) or the existence of infinite splitting universes. If we reject those unobservable overheads, non-determinism is the most direct conclusion.
I'm not a fan of hidden variable theories. Every interpretation asks you to accept something weird. I like the relational interpretation because its weirdness is the least unpalatable to me (no hidden variables, no non-locality, no unobservable "other worlds"). To me the relational interpretation gives an explanation for how certain information can only be predicted with probabilities without requiring some fundamental randomness. The simplified example is that when you measure the spin of an electron the value of the spin only makes sense relative to the thing that measured it. It's essentially just extending to quantum mechanics the idea that the velocity of a thing means nothing without another thing its velocity is relative to.