Why and how is a quantum computer faster than a regular computer?

I'm currently reading a book (and a lot of wikipedia) about quantum physics and I've yet to understand how a quantum computer can be faster than the computers we have today.

How can a quantum computer solve a problem in sub-exponential time that a classic computer can only solve in exponential time?

A quantum computer by itself isn't faster. Instead, it has a different model of computation. In this model, there are algorithms for certain (not all!) problems, which are asymptotically faster than the fastest possible (or fastest known, for some problems) classical algorithms.

I recommend reading The Limits of Quantum by Scott Aaronson: it's a short popular article explaining just what we can expect from quantum computers.

StackExchange Computer Science Q#21727, answer score: 40