$\newcommand{bal}{\begin{align}}\newcommand{eal}{\end{align}}$Bellman's principle of optimality becomes apparent when the following optimization problem
$$
\min_{u_0,u_1,\dots,u_{N-1}} J\left(x_0,\{u_k\}_{k=0}^{N-1}\right)
$$
is decomposed into smaller sub-problems as follows. First, because
$$
\min_{u_0,u_1,\dots,u_{N-1}} J\left(x_0,\{u_k\}_{k=0}^{N-1}\right) = \min_{u_0} \left[ \min_{u_1} \left[ \min_{u_2} \left[ \cdots \left[ \min_{u_{N-1}} J\left(x_0,\{u_k\}_{k=0}^{N-1}\right) \right]\right]\right]\right] \tag{1} \label{eq:decomp}
$$
then we can re-write the optimization problem as
\begin{align}
\min_{u_0} \left[ \min_{u_1} \left[ \min_{u_2} \left[ \cdots \left[ \min_{u_{N-1}} \sum_{k=0}^{N-1} g_k (x_k,u_k) + g_N(x_N) \right]\right]\right]\right]
\end{align}
We can expand the inner summation to get
$$
\min_{u_0} \left[\min_{u_1} \left[\min_{u_2} \left[\cdots \left[\min_{u_{N-1}} g_0 (x_0,u_0) + g_1 (x_1,u_1) + g_2 (x_2,u_2) + \cdots + g_{N-1} (x_{N-1},u_{N-1}) + g_N(x_N) \right]\right]\right]\right]
$$
Because some of the terms in the objective function are constant with respect to the $\min$ operators, then we can take them out as follows (I've broken up the expression over multiple lines for clarity, as it gets too long to fit on one line)
$$
\begin{align}
&\min_{u_0} g_0 (x_0,u_0) \ + \\
&\Big[\min_{u_1} g_1 (x_1,u_1) \ + \\
&\Big[\min_{u_2} g_2 (x_2,u_2) \ + \\
&\Big[\cdots \\
&\Big[\min_{u_{N-2}} g_{N-2}(x_{N-2},u_{N-2}) \ + \\
&\Big[\min_{u_{N-1}} g_{N-1} (x_{N-1},u_{N-1}) \ + \\
&\Big[g_N(x_N)\Big]\Big]\Big]\Big]\Big]\Big]
\end{align}
$$
We then let $J_N(x_N) = g_N(x_N)$, which represents the terminal cost of being in the final state $x_N$. Using this substitution for $g_N(x_N)$, the optimization problem becomes
$$
\begin{align}
&\min_{u_0} g_0 (x_0,u_0) \ + \\
&\Big[\min_{u_1} g_1 (x_1,u_1) \ + \\
&\Big[\min_{u_2} g_2 (x_2,u_2) \ + \\
&\Big[\cdots \\
&\Big[\min_{u_{N-2}} g_{N-2}(x_{N-2},u_{N-2}) \ + \\
&\Big[\min_{u_{N-1}} g_{N-1} (x_{N-1},u_{N-1}) + J_N(x_N)\Big]\Big]\Big]\Big]\Big]
\end{align}
$$
Next, we let
$$
\bal
J_{N-1}(x_{N-1}) &= \min_{u_{N-1}} g_{N-1} (x_{N-1},u_{N-1}) + J_N(x_N)
\eal
$$
Because $x_N = f\left(x_{N-1},u_{N-1}\right)$, then
$$
\bal
J_{N-1}(x_{N-1}) &= \min_{u_{N-1}} g_{N-1} (x_{N-1},u_{N-1}) + J_N\left(f\left(x_{N-1},u_{N-1}\right)\right)
\eal
$$
Note that $J_{N-1}(x_{N-1})$ represents the minimal cost with respect to $u_{N-1}$ of a one-stage process with initial condition $x_{N-1}$. Using this substitution, the optimization problem becomes
$$
\begin{align}
&\min_{u_0} g_0 (x_0,u_0) \ + \\
&\Big[\min_{u_1} g_1 (x_1,u_1) \ + \\
&\Big[\min_{u_2} g_2 (x_2,u_2) \ + \\
&\Big[\cdots \\
&\Big[\min_{u_{N-2}} g_{N-2}(x_{N-2},u_{N-2}) + J_{N-1}(x_{N-1})\Big]\Big]\Big]\Big]
\end{align} \tag{2} \label{eq:opt_prob}
$$
To see where Bellman's principle of optimality comes from, we just need to perform one last substitution:
\begin{align}
J_{N-2}(x_{N-2}) &= \min_{u_{N-2}} g_{N-2}(x_{N-2},u_{N-2}) + J_{N-1}(x_{N-1}) \\
&= \min_{u_{N-2}} g_{N-2}(x_{N-2},u_{N-2}) + J_{N-1}\left(f\left(x_{N-2},u_{N-2}\right)\right)
\end{align}
Here, $J_{N-2}(x_{N-2})$ represents the minimal cost with respect to $u_{N-2}$ of a two-stage process with initial condition $x_{N-2}$. In this context, Bellman's principle of optimality states that, if $J_{N-2}(x_{N-2})$ is minimized with respect to $u_{N-2}$ and $u_{N-1}$, then, regardless of the initial condition $x_{N-2}$ and the chosen control decision $u_{N-2}$, $J_{N-1}(x_{N-1})$ must be minimized with respect to $u_{N-1}$. To see why this must be true, we expand the $J_{N-1}(x_{N-1})$ term in the expression above for $J_{N-2}(x_{N-2})$ to get
\begin{align}
J_{N-2}(x_{N-2}) &= \min_{u_{N-2}} g_{N-2}(x_{N-2},u_{N-2}) + J_{N-1}(x_{N-1}) \\
&= \min_{u_{N-2}} \left[ g_{N-2}(x_{N-2},u_{N-2}) + \left[\min_{u_{N-1}} g_{N-1} (x_{N-1},u_{N-1}) + J_N\left(f\left(x_{N-1},u_{N-1}\right)\right)\right]\right] \tag{3} \label{eq:bellman_part2} \\
&= \min_{u_{N-2},u_{N-1}} g_{N-2}(x_{N-2},u_{N-2}) + g_{N-1} (x_{N-1},u_{N-1}) + J_N\left(f\left(x_{N-1},u_{N-1}\right)\right) \tag{4} \label{eq:bellman_part1}
\end{align}
We can see from \eqref{eq:bellman_part1} that $J_{N-2}(x_{N-2})$ is indeed minimized with respect to $u_{N-2}$ and $u_{N-1}$. Additionally, from \eqref{eq:bellman_part2}, we see that $J_{N-1}(x_{N-1})$ is minimized with respect to $u_{N-1}$, independent of $x_{N-2}$ and $u_{N-2}$. In essence, Bellman's principle of optimality is a result of the property described in \eqref{eq:decomp} and the structure of the objective function $J\left(x_0,\{u_k\}_{k=0}^{N-1}\right)$.
One other important concept to be aware of is that $J_{N-m}(x_{N-m})$ represents the minimal cost of an $m$-stage process with initial condition $x_{N-m}$ and the minimal cost for the final $m$ stages of an $N$-stage process with state at at time-step $k = N-m$ of $x_{N-m}$. In other words, as long as $N \geq m$, then every $m$-stage process is embedded inside an $N$-stage process. This means that, if $\{u_k^*\}_{k=0}^{N-1}$ are the optimal controls for the $N$-stage process with initial condition $x_0$, then $\{u_k^*\}_{k=N-m}^{N-1}$ are the optimal controls for the $m$-stage process with initial condition $x_{N-m}$. This is another way of stating Bellman's principle of optimality.
Going back to the optimization problem in \eqref{eq:opt_prob}, if we proceed with the rest of the substitutions until $k=0$, we get the following recurrence relation $$J_{k}(x_k) = \min_{u_k} \left[g_k(x_k,u_k) + J_{k+1}(x_{k+1})\right]$$ for $k = 0,1,\dots,N-1$ and the boundary condition $J_N(x_N) = g_N(x_N)$, which can be solved recursively starting from the boundary condition.
