Let $\pi$ be a plane in an $d$-dimensional space with normal vector $\underline{w} = [w_1, \dots,w_d]^T$. If point $\underline{p} = [p_1, \dots,p_d]^T$ is in the plane and $\underline{x}= = [x_1, \dots,x_d]^T$ denotes a generic point in this space, we can write the plane equation as
\begin{equation} \pi := \quad \underline{w}^T\cdot (\underline{x} - \underline{p}) = 0. \end{equation}
If we develop this expression and rewrite it with summations we get
\begin{equation} \pi := \quad \sum_{i=1}^d w_ix_i +w_0=0, \end{equation}
where $w_0 = - \sum_{i=1}^d w_ip_i$.
If we chose a different point in the plane, for instance $\underline{q} = [q_1, \dots,q_d]^T$, and proceed as before, we obtain
\begin{equation} \pi := \quad \sum_{i=1}^d w_ix_i +\tilde{w}_0=0, \end{equation}
where $\tilde{w}_0 = - \sum_{i=1}^d w_iq_i$. Thus, the conclusion is that
\begin{equation} \sum_{i=1}^d w_ip_i = \sum_{i=1}^d w_iq_i, \end{equation}
That is, the dot product between the normal vector and any plane point is always the same. I have tried to see this as
\begin{equation} ||\underline{w} || \cdot || \underline{p} || \cdot \cos(\theta_{\underline{w},\underline{p}}) = ||\underline{w} || \cdot || \underline{q} || \cdot \cos(\theta_{\underline{w},\underline{q}}), \end{equation}
which implies that $|| \underline{p} || \cdot \cos(\theta_{\underline{w},\underline{p}}) = \cdot || \underline{q} || \cdot \cos(\theta_{\underline{w},\underline{q}})$, where $\theta_{\underline{a},\underline{b}}$ denotes the angle between two vectors $\underline{a}$ and $\underline{b}$.
Which is the geometric interpretation of this?
Imagine the plane in $d$ space. Then take the unit normal $n$ to the plane and attach it to the origin. The line through the origin with direction $n$ will hit the plane in distance $\delta$ (or $-\delta$, depending on the sign of $n$).
Now if $p$ is any point in the plane and $v$ the vector from the origin to $p$, then $\langle n, v\rangle$ is just the length of the orthogonal projection of $v$ onto the line through the origin with direction $n$. So the plane consists of those points the position vector $v$ of which have the property that the length of the orthogonal projection of $v$ onto this line is just the distance from the plane to the origin.
(This is most easily visualized by looking at a line in $\mathbb{R}^2$ not passing through the origin).