Why does two terms immediately adjacent "mean" multiply? I am currently teaching a GED math class. While learning about the order of operations, the students asked why does a number next to a parentheses mean multiplication?

I understand the rule that two terms next to each mean multiply those two terms together, regardless as to what those terms look like. But did this notation that we now all understand come from shorthand? Or was it easier to read? Or is there some other reason entirely?

Any insight that you can provide will be appreciated. I would get very angry with teachers in my life who explained things by saying, because that is the way it is. If this is the case, then I at least want to be able to tell them that honestly.
I would argue a case for how we describe multiples using natural language. When I say, "I have three boxes", I don't need any other words between 'three' and 'boxes'. Similarly, "I have three $x$", in shorthand, becomes simply $3x$.

Contrast this with addition: "I have an apple and two oranges". The 'and' separates objects that are together but are not necessarily of the same form. "I have an $x$ and two $y$s" becomes $x+2y$.
Teophile's answer is great.

I would like to add that multiplication is the "natural" mathematical operation, not addition, as some people might think. A group has multiplication, but no addition. Addition is only introduced in rings, once we already have a multiplication.

Also, function concatenation is a multiplication, and mathematicians like to write this without an operator (look e.g. into the Lambda calculus), unless we talk about function application. But even then, mathematicians do not need an additional symbol - note how there is no extra symbol on the right hand side.

$$
  (fg)(x) = f(g(x))
$$

(you might more be used to write $(fg)(x)$ as $f\circ g(x)$ ($\circ$ here being function concatenation))