How do the three Miller indices (hkl) denote planes orthogonal to the reciprocal lattice vector?
There are two equivalent ways to define the meaning of the Miller indices: via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors.


The reflecting plane are parallel to a plane that includes the following three points:

$$\frac{\mathbf{a}_1}{h}; \frac{\mathbf{a}_2}{k}; \frac{\mathbf{a}_3}{l}$$

To show that the plane defined by these three points is perpendicular to $h \mathbf{b}_1 + k \mathbf{b}_2 + l\mathbf{b}_3$, we can calculate the dot product between vectors in the plane and the supposedly perpendicular reciprocal lattice vector. To get vectors in the plane, we simply subtract pairs of points in the plane.

$$\frac{\mathbf{a}_1}{h} - \frac{\mathbf{a}_2}{k}$$
$$\frac{\mathbf{a}_1}{h} - \frac{\mathbf{a}_3}{l}$$

Now we multiply it with the reciprocal lattice vector:

$$(\frac{\mathbf{a}_1}{h} - \frac{\mathbf{a}_2}{k}) (h \mathbf{b}_1 + k \mathbf{b}_2 + l\mathbf{b}_3) = \mathbf{a}_1 \mathbf{b}_1 - \mathbf{a}_2 \mathbf{b}_2 = 1 - 1 = 0$$
$$(\frac{\mathbf{a}_1}{h} - \frac{\mathbf{a}_3}{l}) (h \mathbf{b}_1 + k \mathbf{b}_2 + l\mathbf{b}_3) = \mathbf{a}_1 \mathbf{b}_1 - \mathbf{a}_3 \mathbf{b}_3 = 1 - 1 = 0$$

The mixed terms vanish because they are orthogonal. Because the dot products are equal to zero, we have shown that the vector is perpendicular to the reflecting plane.


  [OP] My understanding is that the real (real space) lattice vector is described by the Miller indices as $h{\mathbf  {a_{1}}}+k{\mathbf  {a_{2}}}+\ell {\mathbf  {a_{3}}}$, where the $\mathbf{a}_i$ are linearly independent basis vectors.


No, the significance of $h, k, l$ in defining reflecting planes in terms of the unit cell vectors is as described in the textbook (inverse intercepts along the lattice vectors). 


  [OP] The last point that I'm having a bit of trouble with is how a plane can be defined in such a way that it is orthogonal to three basis vectors


That is not how it is defined. It is defined by the intersections of a plane with the three axes, i.e. through three points that are in the plane.