Since this paper, there have been several different axiomatisations, arguably simpler than this one. For instance in: https://arxiv.org/pdf/2007.13739.pdf and https://arxiv.org/pdf/1812.09114.pdf. In the last one, in particular, all these rules (except (E)) are replaced by:

[![enter image description here][1]][1]

where angles on one side of the equation are bound in a non-trivial way to those on the other side. This rule can be understood as a representation of Euler angles (https://en.wikipedia.org/wiki/Euler_angles).

If you are interested specifically in the axioms you mention, here are some remarks:

The supplementarity rule (SUP) was introduced in https://arxiv.org/pdf/1103.2811.pdf and proven necessary in https://arxiv.org/pdf/1506.03055.pdf. This is an equation that proves useful in the study of the $|W\rangle$ state, expressed as a ZX-diagram.

The rule (E) was introduced in https://arxiv.org/pdf/1702.01945.pdf. In a way, it comes from the fact that $e^{i\frac\pi4}$ can also be written as $\frac{1+i}{\sqrt2}$.

The meaning of rules (C) and (A) are discussed in the paper at Section 7 (p.31).

Up to yanking of the wires and simplification using the spider rules, (C) can be seen as commutation of two unitaries, one that is controlled by a qubit, and the other that is "anti-controlled" by the same qubit.

Decomposing rule (A) the appropriate way, it can be seen as an expression of the following equation : $\begin{pmatrix}1&0&0&0\\0&1&1&0\end{pmatrix}\circ\left(\begin{pmatrix}1\\a\end{pmatrix}\otimes\begin{pmatrix}1\\b\end{pmatrix}\right) = \begin{pmatrix}1\\a+b\end{pmatrix}$. You can find more detail in this Section 7 I mentionned.

Finally there's (BW). To understand this equation, we first need to realise that the diagram:

[![enter image description here][2]][2]

represents the shearing operator $\begin{pmatrix}1&1\\0&1\end{pmatrix}$. It so happens that we can decompose any 2D rotation as a sequence of 3 shear operations along the two canonical axes (see e.g. [this link][3]). Up to the usual ZX-rules for simplification, and up to rearranging of the shear operations, (BW) represents the special case of a rotation by $\pi$:

$\begin{pmatrix}0&1\\-1&0\end{pmatrix} = \begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}1&0\\-1&1\end{pmatrix}\begin{pmatrix}1&1\\0&1\end{pmatrix}$


  [1]: https://i.stack.imgur.com/xxTZq.png
  [2]: https://i.stack.imgur.com/tnn42.png
  [3]: https://www.ocf.berkeley.edu/~fricke/projects/israel/paeth/rotation_by_shearing.html