This is an old-style approach typically used in physics. I shall explain its purpose.
The mathematical part is very easy. We consider a three-dimensional real vector space $V$ with an inner product $\langle -, - \rangle$. In this abstract setting we do not have coordinates for the elements of $V$. However, we can introduce such coordinates by choosing an orthonormal basis $\mathcal B = (b_1, b_2, b_3)$ for $V$ (this means that $\langle b_i, b_j \rangle = \begin{cases} 1 & i = j \\ 0 & i \ne j \end{cases}$). Then each $v \in V$ has a unique representation
$$v = v_1b_1 + v_2b_2 + v_3b_3$$
with $v_i \in \mathbb R$. These numbers are the coordinates of $v$ with respect to $\mathcal B$. The orthonormal basis $\mathcal B$ will also be called a coordinate system for $V$.
Choosing coordinate system means nothing else than choosing an orthogonal linear map $\phi : \mathbb R^3 \to V$ (this means that $\langle \phi(a), \phi(b) \rangle = \langle a, b \rangle$ for all $a, b \in \mathbb R^3$; note that the $\phi(e_i)$ form an orthonormal basis of $V$, where the $e_i$ are the standard basis vectors of $\mathbb R^3$).
If we choose another orthonormal basis $\mathcal B' = (b'_1, b'_2, b'_3)$ for $V$, we get another coordinate representation
$$v = v'_1b'_1 + v'_2b'_2 + v'_3b'_3 .$$
What is the relation between the triples $v_{\mathcal B} = (v_1, v_2, v_3)$ and $v_{\mathcal B'} = (v'_1, v'_2, v'_3)$?
Consider the corresponding orthogonal maps $\phi, \phi' : \mathbb R^3 \to V$. Then $(v'_1, v'_2, v'_3) = (\phi' \circ \phi^{-1})(v_1, v_2, v_3)$. The map $\phi' \circ \phi^{-1}$ is an orthogonal automorphism of $\mathbb R^3$ which can be represented by an orthonormal matrix
$$ A_{\mathcal B, \mathcal B'} =\begin{pmatrix}
l_1 & m_1 & n_1 \\
l_2 & m_2 & n_2 \\
l_3 & m_3 & n_3
\end{pmatrix}$$
Orthonormality means that the six equations in your question are satisfied. We then get
$$\begin{pmatrix}v'_1 \\ v'_2 \\ v'_3\end{pmatrix} = A_{\mathcal B, \mathcal B'} \cdot \begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix} . \tag{1}$$
This gives the three equations in your question. It is the coordinate transformation rule for vectors $v \in V$.
We now come to physics which is the more interesting part.
Physical entities like velocity, electric field, etc. live in our $3$-dimensional space, and they are observed and quantified via certain measurements. Since space does not have a distinguished coordinate system, we have to choose a coordinate system $\mathcal B$ before we make the appropriate measurements to quantify a physical entity $X$. With respect to the chosen coordinate system $\mathcal B$ our measurements produce an $n$-tuple $(x_1,\ldots,x_n)$ of real numbers describing $X$. For example, quantifying a velocity produces triples. But of course there are entities which need more than three measurement components.
Harashima seems to consider only physical entities $X$ which are quantified by triples. The measurement process associates to each coordinate system $\mathcal B$ a triple $X_{\mathcal B} = (x_1, x_2,x_3)$. But is the resulting collection $\{ X_{\mathcal B} \}$ indexed by all orthonormal bases $\mathcal B$ a vector in the above sense, i.e. does it satisfy the coordinate transformation rule
$$X_{\mathcal B'} = A_{\mathcal B, \mathcal B'} \cdot X_{\mathcal B} ?$$
This is not a priori clear. Some physical entities do and are these are called vectors in physics. We could also call them vectorial entities.
Other physical entities do not satisfy the coordinate transformation rule, thus they are no vectors in the sense of physics. Well-known examples are pseudovectors (aka axial vectors) like angular velocity and angular momentum. See here.
Also have a look at Conceptual difference between Covariant and Contravariant tensors.
