DaftWullie's answer outlines the idea pretty well, so I wanted to provide some more specifics on the fundamental gadget he describes and lay it out a bit more methodically.
Once the first step is performed, there exists a quantum state that is distributed over a set of qubits where qubits are at most pairwise entangled. If we now consider simulating the application of an arbitrary set of two and one qubit measurements to this state, then there are six different cases:
- A single-qubit measurement on a single non-entangled qubit.
- A single-qubit measurement on one qubit of a two-qubit entangled state.
- A two-qubit measurement on two disentangled qubits, both of which are non-entangled qubits.
- A two-qubit measurement on an entangled pair of qubits.
- A two-qubit measurement on two disentangled qubits, where only one is part of two-qubit entangled state.
- A two-qubit measurement on two disentangled qubits, where both are part of two separate two-qubit entangled states.
For 1, it is easy to see that this is classically simulable, since there are only two probabilities to calculate. Once these are calculated, the target qubit is no longer relevant and may be ignored by the simulation.
For 2, there are similarly only two probabilities to calculate, except this time, the measurement leaves a single qubit, which may be part of another measurement that can be part of a future instance of cases 1, 3, or 5.
For 3 and 4, since only four states are needed to span a two-qubit measurement basis, there are only four possible probabilities to calculate, regardless of whether the state is entangled or not. Similarly to 1, once these probabilities are calculated, both qubits may be ignored.
For 5, again, only four probabilities need be calculated, but this time, after the measurement, a single qubit remains, which similarly can be part of a future instance of cases 1, 3, or 5.
For 6, we also have four probabilities that need calculating, except now we have the case that the state which exists on the two remaining qubits which are not part of the measurement. This phenomena is known as "entanglement swapping". Hence, in the simulation, we apply the measurement and create a two-qubit entangled state that can then be part of a future instance of cases 2, 4, or 6.
So the algorithm goes as follows:
- Apply the first stage, creating a state with at most pairwise entanglement.
- For each two- or single-qubit measurement in the second stage, extract the desired measurement outcome probabilities and sample any output state accordingly. Because you never create a state which cannot be later simulated efficiently, this process can be repeated until you have performed all measurements.
- Any unmeasured qubits are simply discarded.
Adding a third layer breaks this because now it is possible to create a state which is more than pair-wise entangled on which universal quantum computation can be performed. To see this, observe that such a protocol is equivalent to three stages of pairwise entanglement followed by a round of single-qubit measurements. It can then be seen that three such entangling stages are sufficient to produce the trivalent hexagonal cluster state lattice, a state which is universal for measurement-based quantum computation  (along with single-qubit measurements and classical communications). Once we have such a state, the final stage could then be used to perform a universal measurement-based quantum computation. Hence, if it were possible to simulate such states efficiently, we would be able to simulate quantum computers efficiently as well, which we do not expect to be true.
 Van den Nest, Maarten, et al. "Universal resources for measurement-based quantum computation." Physical review letters 97.15 (2006): 150504. Link here.