Formal definition of a random variable

The way I like to think of it is that it is a function that, in a sense, relieves the problem of dealing with nonnumerical elements by assigning each of them a real number (or real-valued vector) so that they can be compared on the real number line.

For instance, let's say I want to figure out how likely it is for a randomly considered member of the population to be no taller than my bio teacher Jim is. Even though it is not impossible to assign a set whose members fit this criteria working entirely within the sample space of human beings, and then to assign a certain measure of probability $P$ to that set, assigning to each person their height is a function that makes this task a bit easier.

Numerically speaking, let $X$ be the function from the sample space of human beings to the (nonnegative) real line that assigns to each person $\omega$ a height $X(\omega)$. Let's say Jim's height is 6 (feet). When we say, then, in layman's terms, what percentile Jim's height represents, what we mean to do is to ask for $P\{\omega : X(\omega) \leq X(Jim)\}$, which is the measure of the set of people (i.e. the probability measuring of the set of people) whose heights do not exceed Jim's. Notice that the function sends us to a nice, ordered place where just considering the people themselves without this numerical value would not be sufficient for this task.

A random variable just a fancy name for a measurable function on a probability space $(\Omega, \cal{F}, P)$. When we write $\{X \in S\}$, this is just notational shorthand for $\{ \omega \in \Omega : X(\omega) \in S\}$.

If you don't see this, I recommend you read a basic measure theoretic probability book, like Durrett's Probability: Theory and Examples, where Random Variables are defined.