In view of the context, no computer is involved. Rather, one asks for a method to generate a random variable $X$ with a given distribution (in your case, Gamma) from a collection, possibly infinite, of independent random variables, which are either uniformly distributed over $(0,1)$, or some simple transformations of these.
Assume for example that one asks for a standard gaussian random variable. A popular approach, called the Box-Muller method, is to use $U_1$ and $U_2$ i.i.d. uniform on $(0,1)$ and to set $X=\sqrt{-2\log U_1}\cos(2\pi U_2)$. See here for more explanations and examples.
In the Box-Muller method, one uses $U_1$ and $U_2$ to get one random variable $X$. Acception-rejection methods are different because they use a random number of random variables to get $X$. Their general principle is to use a given number of i.i.d. random variables, say two uniform random variables $U_1$ and $U_2$, and to test if these satisfy a well chosen property, say $\mathcal Q(U_1,U_2)$. If $\mathcal Q(U_1,U_2)$ holds, one accepts $(U_1,U_2)$ in the sense that one sets $X=\Phi(U_1,U_2)$ for a suitable function $\Phi$ and the game is over. Otherwise, that is, if $\mathcal Q(U_1,U_2)$ does not hold, one rejects $U_1$ and $U_2$, and one performs the same test with two other uniform random variables $U_3$ and $U_4$, say. If $\mathcal Q(U_3,U_4)$ holds, one accepts $(U_3,U_4)$ in the sense that one sets $X=\Phi(U_3,U_4)$ and the game is over. Otherwise one rejects $U_3$ and $U_4$ and one turns to $U_5$ and $U_6$, and so on. One can also keep some information from the rejected random values, for example their number, to make the value $X$, but the procedure above is always the rough idea.
Hence, in rejection methods, the total number of random variables $U_n$ used to produce one value of $X$ is random. In your case, you are given a possibly infinite sequence $(U_n)_{n\geqslant1}$ of i.i.d. uniform random variables and you are asked to devise an algorithm to produce one random variable $X$ with the desired Gamma distribution, using almost surely a finite number of random variables $U_n$.
A good and relatively short introduction is the first chapter (The acceptance rejection method with applications; generating a standard normal random variable) of the lecture notes of this course. Another source, written for high energy astrophysicists and remarkably down-to-earth, is here.
The second link provides the code of an algorithm to generate Gamma distributions from what they call a Lorentzian distribution, which is also known as a standard Cauchy distribution. This is quite convenient since the latter is simply the distribution of $\tan(\pi U)$ with $U$ uniform on $(-1,1)$. I would rather leave you the pleasure of putting these bits and pieces together to concoct your algorithm, unless you get stuck at some point.