The key observation is that cosine and sine are bijections, i.e. one-to-one functions, between the set $[0,1]$, where $|\alpha|$ and $|\beta|$ reside, and the set $[0,\frac{\pi}{2}]$, where $\frac{\theta}{2}$ resides.
Bijection argument
First, write $\alpha$ and $\beta$ in polar form
$$
\alpha=re^{ia}\quad\beta=se^{ib}\tag1
$$
with $r,s\in[0,1]$ and $a,b\in[0,2\pi)$ and note that $(1.3)$ imposes no constraints on the phase angles $a$ and $b$. In fact, we can just set $\gamma:=a$ and $\varphi:=b-a$. However, equation $(1.3)$ does constrain the absolute values $|\alpha|=r$ and $|\beta|=s$ by prescribing that
$$
r=\cos\frac{\theta}{2}\quad s=\sin\frac{\theta}{2}\tag2
$$
for some $\theta\in[0,\pi]$. Now, cosine is a bijection between $[0,\frac{\pi}{2}]$ and $[0,1]$, so for any $r\in[0,1]$ there exists a unique $\theta\in[0,\pi]$ such that $r=\cos\frac{\theta}{2}$. Then $s\in[0,1]$ is uniquely determined from $s=\sqrt{1-r^2}$. Moreover,
$$
s=\sqrt{1-r^2}=\sqrt{1-\cos^2\frac{\theta}{2}}=\sin\frac{\theta}{2}\tag3
$$
as expected.
Geometric argument
The above argument can be given geometric interpretation by treating the pair $(r, s)$ as coordinates of a point in the first quadrant of a Cartesian coordinate plane. The points that satisfy the requirement $$r^2+s^2=|\alpha|^2+|\beta|^2=1\tag4$$ lie on the unit circle with the center at the origin. Instead of identifying a point using $(r,s)$ coordinates, we can identify it by specifying the polar angle $\phi\in[0,\frac{\pi}{2}]$ which is related to $(r,s)$ by
$$
r=\cos\phi\quad s=\sin\phi.\tag5
$$
By setting $\theta:=2\phi$ we recover $(2)$ .
