General solution of linear equation = particular + general homogeneous solution 
Let $x_p$ be the particular solution of $Ax = b$ and $x_h$ be the solution to
the homogeneous system $Ax = O$. All the solutions of $Ax = b$ are of the form $x_p + x_h$

Proof:

Let $x$ be the solution of $Ax = b$, then $A(x − x_p) = Ax − Ax_p = b − b = 0 \to x_h = x − x_p \to x = x_p + x_h$
We need to show all the solutions are of this format $x_p + x_h$.
Let $x'$ be a solution of $Ax = 0$, then
$A(x + x') = Ax + Ax' = Ax + 0 = b + 0 = b$.
Hence $x + x'$ is a solution of $Ax = b$.

I understand the technical details of this proof, but I am not sure about the intent of the arguments.
I think the first part is saying if $x $ is the solution of $Ax = b$, then $x = x_p + x_h.$
Second part is saying that $x_p + x_h$ is a solution to every system $Ax= b.$
Does that make sense?

Yes, presented more clearly
$$\overbrace{{\rm if}\ \  ax_p = b}^{{\rm particular\ solution\ }{\large x_p}}\!\! {\rm then}\  \ ax=b \!\iff\! a(x\!-\!x_p) = 0\!\iff\! \ \underbrace{x-x_p = x_h\ \  {\rm and} \overbrace{ a\, x_h = 0}^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\rm homogeneous\ solution}\ \large x_h}}_{\large \bbox[5px,border:1px solid #c00]{\begin{align}x\ &=\ \text{general solution }\\ &=\ \rm particular + homogeneous\end{align}}}\qquad$$
Remark $\ $ More generally this hold for any  operator $A$  that is  $\,\rm\color{#c00}{L}inear,\,$ i.e.
$\quad $ if $\,\ \color{#0a0}{Ax_p = b}\,\ $ then $\,\ Ax=b \!\iff\! A(x\!-\!x_p) = 0\!\iff\! \ x= x_p + x_h\,$ and $\, Ax_h = 0$
because  $\ \ \ \smash[t]{ A(x-x_p)\overset{\rm\color{#c00}{\large L}} = Ax - \color{#0a0}{Ax_p} = Ax\! -\!\color{#0a0}b} $
therefore $\ A(x-x_p)\, =\, 0 \ \ \iff\ \  Ax = b$
Thus this relationship between general, particular and homogeneous solutions also holds for linear differential and difference equations (recurrences), linear Diophantine equations (Bezout scalings), and many other common linear equations. In linear algbra such solutions spaces of inhomogenous equations are abstracted in the study of affine spaces.
