I don't understand the final bit - how does entanglement allow scientists to know the value of the qubits? I can understand how it could allow them to perform complicated calculations without measuring the qubits, so they can remain in superposition throughout the operation. But how can they know the value of them through entanglement?

I think the article is referring to quantum error correction, which does use highly entangled states. It is not strictly correct to say that:

Instead, quantum information is encoded in a subspace of the whole Hilbert space, so that the states |0> and |1> are actually encoded in highly entangled states. A simple example of this is the 9-qubit code found by Shor, in which

The encoding of a general state a|0> + b|1> is such that if any one of the nine qubits suffer a bit flip error (|0> -> |1>, |1> -> |0>) or a phase flip error
(|0> -> |0>, |1> -> -|1>) or a combination of the two, then the error can be corrected by making measurements on the encoded state that reveal the error without telling you anything about the state. Thus, the error can be corrected without collapsing the superposition.

It can be shown that this is sufficient to correct a much more general class of errors than just bit flips and phase flips.