Nielsen's paper cited in the question simplifies the arguments originally laid out in two papers by Horodecki family. This answer sketches the original arguments and is meant to complement the nice explanation based on representation theory written by @Markus Heinrich by requiring less background knowledge and hopefully providing some additional insight into the relationship between depolarizing channels and twirling. It also demonstrates the use of state-channel duality.
High level summary
The argument uses state-channel duality to translate twirling of channels to $U\otimes U^*$ twirling of states. By unitary invariance of the Haar measure, twirling is idempotent, so the Choi matrix of a twirled channel is invariant under $U\otimes U^*$ twirling of states. However, it turns out that the only states invariant under $U\otimes U^*$ twirling of states are the so-called noisy singlets which under state-channel duality correspond to depolarizing channels.
Noisy singlet
Consider two systems with the Hilbert spaces of the same finite dimension $N$. Let $|\psi\rangle:=\frac{1}{\sqrt{N}}\sum_{i=1}^N|i\rangle|i\rangle$. It is easy to check that for any linear operator $A$
$$
(A\otimes I)|\psi\rangle = (I\otimes A^T)|\psi\rangle.\tag1
$$
Now, for $p\in[0,1]$, we define the noisy singlet $\rho_p$ to be the bipartite state
$$
\rho_p:=p|\psi\rangle\langle\psi|+(1-p)\frac{I\otimes I}{N^2}.\tag2
$$
Twirling
Twirling of states sends a bipartite state $\rho$ to
$$
\rho_t := \int dU (U\otimes U^*)\rho(U^\dagger\otimes U^T)\tag3
$$
where $U^*$ denotes the complex conjugate of $U$. Using $(1)$, we can show that the Choi matrix $J(\hat\Lambda_t)$ of a twirled channel $\hat\Lambda_t$ is the result of twirling of states applied to the Choi matrix $J(\hat\Lambda)$ of the original channel $\hat\Lambda$
$$
\begin{align}
J(\hat\Lambda_t)&=\hat\Lambda_t\otimes\hat{I}(N|\psi\rangle\langle\psi|)\\
&=\left(\int dU\hat{U}\circ\hat{\Lambda}\circ\hat{U}^\dagger\right)\otimes\hat{I}(N|\psi\rangle\langle\psi|)\\
&=\left(\int dU(\hat{U}\otimes\hat{I})\circ(\hat{\Lambda}\otimes\hat{I})\circ(\hat{U}^\dagger\otimes\hat{I})\right)(N|\psi\rangle\langle\psi|)\\
&=\int dU(\hat{U}\otimes\hat{I})\circ(\hat{\Lambda}\otimes\hat{I})\left((U^\dagger\otimes I)N|\psi\rangle\langle\psi|(U\otimes I)\right)\\
&=\int dU(U\otimes I)\left[\hat{\Lambda}\otimes\hat{I}\left((U^\dagger\otimes I)N|\psi\rangle\langle\psi|(U\otimes I)\right)\right](U^\dagger\otimes I)\\
&=\int dU(U\otimes I)\left[\hat{\Lambda}\otimes\hat{I}\left((I\otimes U^*)N|\psi\rangle\langle\psi|(I\otimes U^T)\right)\right](U^\dagger\otimes I)\\
&=\int dU(U\otimes U^*)\left[\hat{\Lambda}\otimes\hat{I}\left(N|\psi\rangle\langle\psi|\right)\right](U^\dagger\otimes U^T)\\
&=\int dU(U\otimes U^*)J(\hat\Lambda)(U^\dagger\otimes U^T)\\
&=J(\hat\Lambda)_t.
\end{align}\tag4
$$
Another fact we can easily prove using $(1)$ is that every noisy singlet $(2)$ is invariant under twirling of states $\rho_{p,t}=\rho_p$. In fact, it turns out that noisy singlets are the only states with this property. See section $V$ in this paper for a proof of this fact.
Depolarizing channel
Depolarizing channel is a CPTP map defined by
$$
\hat\Delta_p(\rho) = p\rho + (1-p)\frac{I}{N}\mathrm{tr}\rho.\tag5
$$
A short calculation shows that the Choi matrix of $\hat\Delta_p$ is
$$
J(\hat\Delta_p)=(\hat\Delta_p\otimes\hat I)(N|\psi\rangle\langle\psi|)=N\rho_p\tag6
$$
where $\rho_p$ is a noisy singlet.
Putting it all together
Finally, unitary invariance of the Haar measure implies that twirling a channel twice yields the same result as twirling it once
$$
(\hat\Lambda_t)_t=\hat\Lambda_t.\tag7
$$
Therefore, by $(4)$
$$
J(\hat\Lambda_t)=J((\hat\Lambda_t)_t)=J(\hat\Lambda_t)_t\tag8
$$
i.e. the Choi matrix of $\hat\Lambda_t$ is invariant under twirling of states. But noisy singlets are the only states with this property. Therefore, $J(\hat\Lambda_t)$ is (a scalar multiple of) a noisy singlet
$$
J(\hat\Lambda_t)=N\rho_p\tag9
$$
for some $p\in[0,1]$. However, $N\rho_p=J(\hat\Delta_p)$, so by injectivity of $J$, we have
$$
\hat\Lambda_t=\hat\Delta_p\tag{10}
$$
which was to be proven.
