Let $C_i$ be a cluster and $ \mu_i = |C_i|^{-1}\sum_{x\in C_i} x $. Then the point-cost of placing a point $y$ in cluster $j$ is given by:$$ \xi_j(y) = \sum_{x\in C_j} || x - y ||_2^2 $$
We can also define the cost of cluster $ j $ via: $ \eta_j = \xi_j(\mu_j) $.
First, note that
$$ \xi_j(z) = \eta_j + N_j\,|| \mu_j - z ||_2^2 \tag{1} $$
where $N_j = |C_j|$. Why? Because
\begin{align*}
\xi_j(z) - \eta_j
&= \sum_{x\in C_j} ||x-\mu_j||_2^2 - ||x-\mu_j||_2^2\\
&= \sum_{x\in C_j}\sum_k z_k^2 - \mu_{jk}^2 - 2x_kz_k + 2x_k\mu_{jk}\\
&= \sum_k N_jz_k^2 - N_j\mu^2_{jk} - 2z_kN_j\mu_{jk} +2\mu_{jk}^2N_j\\
&= N_j||\mu_j - z||_2^2
\end{align*}
Next consider the merge cost $\Delta_{ij}$, which is the cost of merging clusters $C_i$ and $C_j$. Let $C_{v} = C_i \cup C_j$. Then
$$
\Delta_{ij} = \eta_v - \eta_i -\eta_j
$$
Using equation(1) and $\mu = [N_i\mu_i + N_j\mu_j]/(N_i + N_j)$, we get that:
\begin{align*}
\eta_v
&= \xi_i(\mu_v) + \xi_j(\mu_v) \\
&= \eta_i + N_i||\mu_i - \mu_v||_2^2 + \eta_j + N_j||\mu_j - \mu_v||_2^2\\
&= \eta_i + \eta_j + \frac{N_jN_i^2}{(N_j + N_i)^2}||\mu_i - \mu_j||_2^2 + \frac{N_iN_j^2}{(N_j + N_i)^2}||\mu_j - \mu_i||_2^2\\
&= \eta_i + \eta_j + \frac{N_iN_j}{N_j + N_i}||\mu_j - \mu_i||_2^2 \\
\therefore\; \Delta_{ij} &= \frac{N_iN_j}{N_j + N_i}||\mu_j - \mu_i||_2^2 \tag{2}
\end{align*}
We next want to look at the cost of moving a point, i.e. reassigning from one cluster to another. Let $p\in C_i$. Then then change in cost of reassignment of $p$ from $C_i$ to $C_j$ is:
$$
\Theta_{ij}(p) = \eta_w + \eta_u - \eta_{i} - \eta_j
$$
where $C_w = C_i \setminus p$, $C_p = \{p\}$, and $C_u = C_j \cup C_p$.
Notice that:
$$
\eta_u - \eta_j = \Delta_{jp} = \frac{N_j}{N_j + 1}||\mu_j - p||_2^2
$$
using $\eta_p=0$.
Further, note that
$$
\eta_i - \eta_w = \Delta_{wp} = \frac{N_i-1}{N_i}||\mu_w - p||_2^2
$$
using $C_i = C_w \cup C_p$.
Next, use $\mu_w = (N_i\mu_i - p)/(N_i -1)$ to get
$$
\frac{N_i - 1}{N_i}||\mu_w-p||_2^2 = \frac{1}{(N_i - 1)N_i}||N_i\mu_i-p-(N_i-1)p||_2^2 = \frac{N_i}{N_i - 1}|| \mu_i-p ||_2^2
$$
Clearly, we want to perform a reassignment only if the cost of reassignment is negative. (Meaning the cost was higher before the reassignment). That is, $\Theta_{ij}(p) < 0$.
But we have
\begin{align*}
\Theta_{ij}(p) &= (\eta_u - \eta_j) - (\eta_i - \eta_w) < 0 \\
\implies\;&\; \eta_u - \eta_j < \eta_i - \eta_w \\
\therefore\;&\; \frac{N_j}{N_j + 1}||\mu_j - p||_2^2 < \frac{N_i}{N_i - 1}|| \mu_i-p ||_2^2
\end{align*}
when an improvement occurs, as required.
See Telgarsky and Vattani, Hartigan’s Method: k-means Clustering without Voronoi.
