What is "sampling from a distribution"? Exercise 4.11.3 of Grimmett and Stirzaker's Probability and Random Processes reads "Use the rejection method to sample from the gamma density $\Gamma(\lambda,t)$ where $t (\geq 1)$ may not be assumed integral."

What exactly is this exercise asking for? More generally, what is meant by "sampling from a distribution" (or "sampling from a density")? Section 4.11 of the given textbook refers to "sampling" over and over, but doesn't define it. 

It seems, based on the examples, that it means to simply find a random variable with the given distribution. If that's correct, why is it called "sampling"?

Am I supposed to use a computer for this exercise?
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.
I believe that you would need to use something like MATLAB for that. You need to input the proper values and you can generate a set of random values that you can work from.