How it works depends on the choice of quantum system used for computation. For any choice of quantum system, the common theme is that $\text{CNOT}$ does not collapse the wavefunction, i.e. force a choice between $\vert 0 \rangle$ and $\vert 1 \rangle$, while a measurement does.
A simple example (oversimplified here) uses a non-linear Kerr medium to create a $\text{CNOT}$ gate with two photons acting as qubits. In this case a Hadamard gate ($H$) is created with phase shifters (slabs of transparent media with index of refraction $\ne 1$) and beam splitters (partially silvered glass), which produce the superposition of states.
The Kerr effect is a change in refractive index based on the presence of an electric field in the Kerr medium, and when two photons pass through a Kerr medium they can experience cross-phase modulation. In other words the atoms in the Kerr medium mediate an interaction between the two photons (qubits).
The upshot is that the system can be tuned such that the Kerr medium acts as the gate
$$K = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}.$$
With access to $K$ and $H$, the $\text{CNOT}$ gate ($U_c$) is simply
$$U_c = (I \otimes H) K (I \otimes H) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}.$$
In this manner the $U_c$ gate is implemented without collapsing the wavefunction. Conversely, when a photon interacts with a photon detector (measurement) it is absorbed and converted to current or voltage, collapsing the wavefunction and forcing it to choose one definite state.
As noted above, this is an oversimplified explanation. Since you already have Nielsen and Chuang, you can see a much more rigorous treatment of this example in Section 7.4.2, as well as constructions of $\text{CNOT}$ in the context of ion traps (7.6.3) and nuclear magnetic resonance (7.7.3).
