That interpretation provides an intuitive way to reason about quantum mechanics and therefore makes it much easier to learn how quantum mechanics works, without getting distracted by paradoxes and philosophical difficulties.
To learn more,
Read a paper by one of decoherent histories' main proponents http://arxiv.org/abs/quant-ph/0612065
See if you can understand quantum decoherence: http://en.wikipedia.org/wiki/Quantum_decoherence
Read the FAQ: http://quantum.phys.cmu.edu/CHS/quest.html
Dive into a textbook explaining quantum physics in decoherent histories interpration (by the same proponent as above) http://quantum.phys.cmu.edu/CQT/index.html
And of course its Wikipedia page may not be overlooked: http://en.wikipedia.org/wiki/Consistent_histories