The understanding of random variables and their arithmetic, can't wrap my head around it. Recently started studying probability and statistics. What I first learned is that random variables are like functions taking outcomes from an experiment and plotting them to a number on the real-axis. 
I feel that my interpretation can't be applied to for example the definition of the central limit theorem where we are asked to sum independent identically distributed random variables. Should I think of the addition like the addition of two arbitrary results of the random variables?
 A: You can think of it in the same way you add functions in $\mathbb R\to\mathbb R$, i.e. pointwise operations. If you had functions $f:\mathbb R\to\mathbb R, x\mapsto 2x$ and $g:\mathbb R\to\mathbb R, x\mapsto x+3$, then you would have $(f+g):\mathbb R\to\mathbb R, x\mapsto (2x) + (x+3) = 3x+3$. 
Similarly, if you can think of the sum of two random variables $X$ and $Y$ as $(X+Y):\Omega\to\mathbb R, \omega\mapsto X(\omega)+Y(\omega)$. 
For example, let's say you rolled two die. The sample space would be $\Omega = \{(x,y) : 1\leq x,y\leq 6\}$. If we let $X$ be the first roll and $Y$ be the second roll, then $X:\Omega\to\mathbb R, (x,y)\mapsto x$ and $Y:\Omega\to\mathbb R, (x,y)\mapsto y$. Then the random variable for their sum would be the function $(X+Y):\Omega\to\mathbb R, (x,y)\mapsto X(x,y) + Y(x,y) = x+y$. 
A: Notice the s at the end of the phrase "identically distributed random variables."
The things you add together in your sums for the Central Limit Theorem (CLT)
are not multiple values of one random variable:
they are multiple variables, each of which is a different function from each of the others.
For example, one of the random variables, $X_1,$ might have a distribution
such that its value is greater than $5$ on half of the probability space, where our measurement of what is "half" of the space is made according to probability. Since the next variable, $X_2,$ is identically distributed to $X_1,$ it also is greater than $5$ on half of the probability space,
but not the same half of the probability space.
(If $X_2$ were greater than $5$ whenever $X_1$ is, and only when $X_1$ is, then the two random variables would no longer be independent.)
In order to apply your definition of a random variable (relating it to an experiment) to the CLT, you have to regard the "experiment" as the entire sequence of observations that are averaged together by the CLT, and not as a procedure (yielding one observation) that you can repeat over and over to get a sequence of observations.
To put the example above in more usual terms, sometimes $X_1 > 5$ and $X_2 > 5,$ but sometimes $X_1 > 5$ and $X_2 \leq 5.$
"Sometimes" means "for some outcomes of the experiment."
The experiment has some outcomes for which $X_1$ and $X_2$ produce the same value, but it also has outcomes for which $X_1 > 5$ and $X_2 \leq 5;$
for those outcomes $X_1$ and $X_2$ do not produce the same value,
and therefore each of $X_1$ and $X_2$ is a different function of the outcomes.

Here's a simpler example: toss two coins. Suppose the variable $X_1$ is $1$ if the first coin is heads, $0$ if the coin is tails, while $X_2$ is $1$ or $0$ if the second coin is heads, $0$ if that coin is tails.
The outcomes in the probability space of this experiment can be partitioned into four subsets. Choose any one of the subsets, and the probability that the observed outcome will be in that subset is $\frac14.$
On exactly two of these subsets of outcomes, $X_1 = 1,$ so $P(X_1 = 1) = \frac12.$
Also on exactly two of these subsets, $X_2 = 1,$ so $P(X_2 = 1) = \frac12.$
Similarly, $P(X_1=0) = P(X_2 = 0) = \frac12.$
So the probability distributions of $X_1$ and $X_2$ are the same.
But not every outcome that makes $X_1 = 1$ will make $X_2 = 1.$
In fact, $P(X_1 = 1 \land X_2 = 0) = \frac14.$
