As far as we believe, a quantum Turing machine is able to simulate any quantum computer, and it is also equivalent to classical deterministic Turing machine in terms of computability. In other words, as far as we know the space of problems solvable by quantum computers is the same as space of problems solvable by classical computers.
However, if we consider practical computability, things may look a bit different. Imagine a problem where we have a classical solution which runs with $O(2^n)$ complexity. It is definitely solvable, but for any reasonable data size it will require massive amounts of operations. In practice, it will run for thousands of years, even on the fastest computers. Now imagine we have a quantum algorithm solving the same problem, but with $O(n)$ complexity. Out of a sudden, exact same problem can be solved in minutes, which is very reasonable (especially compared to thousands of years).
I think the original quote could be extended by adding "in any reasonable time", and we get a fair statement.