If you understand Turing Machines, you probably also understand other automata. So you probably understand nondeterministic automata [1].
A quantum computer is like a very restricted nondeterministic automaton, except that the "do several things a once" is implemented in physics. That means just like a NFA can be exponential faster than a DFA, a QC can be exponential faster than a normal computer. But the restriction on QCs makes that a lot harder to do, and so far it only works for some algorithms.
As to why quantum physics allows some kind of nondeterminism: If you look at particles as waves, instead of a single location you get a probability function that tells you "where the particle is". So a particule can be "in several places at once". In the same way a qbit can have "several states at once".
> What I don't understand is how a programmer is supposed to determine the correct solution when their computer is out in some crazy multiverse.
Because one way to explain quantum physics is to say that the waveform can "collapse" [2] and produce a single result, as least as far as the observers are concerned. There are other interpretations of this effect, and this effect is what makes quantum physics counterintuitive and hard to understand.
[1] https://en.wikipedia.org/wiki/Nondeterministic_finite_automa...