[An earlier version of the question asked for an answer that completely avoided mathematics; this answer was an attempt to give some intuitive motivation, at a similar level to the document being asked about.]
The linked page is wrong when it says that $X+X\neq 2X$.
In the example $X$ is a random variable represents the number showing on the face of a die -- the result of an experiment like "roll a six-sided die once and record at the number on the face of the die."
So you roll a die and write down what you saw. Whatever number you would record is $X$... so $X+X$ represents the result added to itself. If you roll another die, that number you would have written down before doesn't change.
Later on the page it says:
When two dice are rolled, though, the results are different. Call the random variable that represents the outcomes of the two-dice process $T$ (for "two"). We could write $T = X + X$. This equation represents the fact that $T$ is the result of two independent instances of the random variable $T$
The very end of that quote is presumably a typographical error, they mean $X$ not $T$ there (since if it was $T$ they just said $T$ was the result of two instances of itself). But with that replacement it's still incorrect.
If you have two independent instances of the experiment (roll a die, record the number showing) you're dealing with two different random variables.
So imagine I have a red die and a blue die. Then I can say "Let the result on the red die be $X_1$ and the result on the blue die be $X_2$". Then we can follow the example at that linked page by defining $T$ to be the sum of the numbers showing on those two dice, so $T=X_1+X_2$. If the dice and the die-rolling process is fair then the distribution of $X_1$ and $X_2$ are the same, but $X_1$ and $X_2$ -- the random variables -- are distinct.
[There's an excellent discussion by whuber of random variables (and sums of them) here, and the concept of random variables is covered in slightly more detail (if in places more technical) here. I recommend you at least read the answer at the first link.]
This problem has come because the author has confused the random variable with its distribution. You can see that here:
In this case, students do think of the random variable X as representing a single, unknown value, in the same way that they think about algebraic variables. But X really refers to the distribution of possible values and the associated probabilities.
He explicitly conflates the random variable with its distribution.
In fact random variables are in many ways just like other algebraic variables and may often be manipulated in the same manner. In particular, a single univariate random variable doesn't stand for two distinct quantities (like the outcome from two different die rolls) at the same time. $X+X$ really is $2X$.