The statement is either false or unproven, depending on how you take it.
Scenario 1: "Quantum Computers Cannot Be Simulated, Even Given Infinite Time and Space"
This statement would be one in which we wish to know the outcome of a quantum computation, but we lack a quantum computer. In this case, is there anyway we can make a classical computer do the computation for us, but maybe it would take much longer? In fact we can. Our quantum computer must have some algorithm, given as a series of gates, which can be written as unitary operation $U$ on the initial state $\left| \psi \right\rangle$. So the quantum computation can be duplicated if we calculate the final state, $U \left| \psi \right\rangle$. This is very hard to do in general--it requires computing an $2^n \times 2^n$ dimensional matrix times a $2^n$ dimensional vector. But it's not impossible, and given lots of time and space you can do it--there's nothing magic about vector multiplication.
(A more sophisticated argument using path-integral analogy can show that quantum computing is possible in polynomial space but exponential time, actually, but let's not worry about it.)
Scenario 2: "Quantum Computers Cannot Be Efficiently Simulated"
This question is slightly different. The question is: suppose a quantum computer can solve a problem in polynomial time--so if it takes $n$ input or output qubits, the runtime is proportional to $n^k$, where $k$ is some finite number. Then does there exist a classical algorithm that does the same problem, although maybe with a different $k$?
Nobody knows. Seriously. We know that there are quantum examples of better $k$'s, as in Grover's algorithm (classical: $k = 1$, quantum: $k = 1/2$). We know that there are quantum polynomial algorithms to solve problems that appear to be exponential in classical computers (factoring; the hidden subgroup problem in general). That said, there is no known proof that a quantum computer can generically do things in polynomial time that a classical computer cannot do. Factoring, for instance, seems to be hard for classical computers, but there is no proof that it definitely is. Many people believe this is so, but complexity theory is a very tricky thing and the logical proof is not there. Perhaps all the problems that look easy for quantum are also easy for classical, and we just haven't figured them out yet. Note that LOTS of time and research is dedicated to running simulations of quantum systems, and it appears to be very difficult to simulate a quantum many-body system. But perhaps we're just not good at it.
In complexity-theory language, we have not yet proven a separation between $BQP$ ("bounded-error quantum polynomial") and $BPP$ ("bounded-error probabilistic polynomial"), the complexity classes of problems efficiently solvable on a quantum and classical computer.