What is the proof that any given unitary matrix can be converted as above?
Let $U$ be an arbitrary $2\times 2$ unitary matrix. This is equivalent to the rows/columns of $U$ being orthonormal bases.
Let us write a generic $U$ as
$$U=\begin{pmatrix}a&b\\c&d\end{pmatrix}.$$
The constraints imposed on the coefficients $a,b,c,d$ by the requirement of $U$ being unitary are
$$|a|^2+|c|^2=1,\qquad |b|^2+|d|^2=1,\qquad a^* b+c^* d=0.$$
A pair of complex numbers $a,b\in\mathbb C$ satisfying $|a|^2+|b|^2=1$ can always be parametrized as
$$a=e^{i\alpha_{11}}\cos\theta,\qquad b=e^{i\alpha_{12}}\sin\theta,$$
for some real coefficients $\alpha_{ij},\theta\in\mathbb R$.
It follows that using only the normalization constraint (but without taking into account the orthogonality) we can parametrize $U$ as
$$U=\begin{pmatrix}e^{\alpha_{11}}\cos\theta& e^{\alpha_{12}}\sin\theta\\
e^{\alpha_{21}}\sin\theta & e^{\alpha_{22}}\cos\theta
\end{pmatrix}.$$
Requiring the columns to be orthogonal adds the constraint
$$e^{i(\alpha_{11}-\alpha_{12})}+e^{i(\alpha_{21}-\alpha_{22})}=0,$$
that is, $\alpha_{11}=\alpha_{12}+\alpha_{21}-\alpha_{22}+\pi$.
We conclude that $U$ is parametrized by four real parameters, here denoted $\theta,\alpha_{12},\alpha_{21},\alpha_{22}$.
To get the form you show you simply need to change variables as follows:
\begin{align}
\theta&=\gamma/2, \\
\alpha_{12} &= \alpha-\beta/2+\delta/2+\pi,\\
\alpha_{21} &= \alpha+\beta/2-\delta/2, \\
\alpha_{22} &= \alpha+\beta/2+\delta/2.
\end{align}
Like why there should be only 4 variables and not more or less?
While the above already proved this, it can be useful to know that this a special case of a more general result. Namely:
a generic unitary $n\times n$ matrix is specified by $n^2$ real parameters (see wiki page on the unitary group for more details).
An easy way to see this is again to remember that unitary matrices are characterised by their columns/rows forming an orthonormal system.
This amounts to $n$ real constraints (imposing each of the $n$ columns to be normalized), plus $\binom{n}{2}=n(n-1)/2$ additional complex constraints (imposing each pair of columns to be orthogonal). Each complex constraint amounts to two real constraints, so this sums up to a total of
$$n+2\binom{n}{2}=n+n(n-1)=n^2$$
real, independent constraints.
A generic $n\times n$ matrix is characterised by $n^2$ complex numbers, that is, $2n^2$ real numbers.
We conclude that the number of free parameters of a generic $n\times n$ unitary matrix is:
$$2n^2-n^2=n^2.$$
Going back to the simple $2\times 2$ case above, you can see how we get back the previous result because $2^2=4$ (you might also notice that to count parameters in the special case $2\times 2$ I used a different ad-hoc strategy, rather then the one showed here to get the count in the general case).
Yet another way to count parameters
Another method I like is to think entirely in terms of orthonormal systems.
The question is: how many parameters need to be given to specify an orthonormal basis in an $n$-dimensional complex vector space?
Let us start by the first vector. The only constraint here is that we want the vector to be normalized.
The number of real parameters needed to specify a normalized vector in $n$-dimensions is $d_1=2n-1$.
Let us now add another vector. Now we have to impose both the normalization of this additional vector (one real constraint), and the orthogonality of this additional vector to the initial one (two real constraints).
The additional parameters are therefore $d_2= 2n-3$.
A third vector will need to be normalized (one real constraint), and orthogonal to the first two vectors ($2\times 2$ real constraints), thus $d_3=2n-5$.
Iterate this reasoning until you get to the last vector, which will be specified by a single real parameter.
The total number of parameters is therefore:
$$\sum_k d_k=(2n-1)+(2n-3)+\cdots+3+1=\sum_{k=0}^{n-1} (2k+1).$$
In other words, the number of parameters is given by the sum of the first $n$ odd integers, which is again readily shown to equal $n^2$.
