Intuitively, What does it mean when you multiply numbers? I asked my professor about what does it mean when we multiply $5\cdot 5\cdot 5$. He said there is no definition of this thing in mathematics.
When I was learning physics in high school I learned about gravitational formula: $$F = G\frac{m_1m_2}{r^2} $$ Since then I was wondering what does it mean when we multiply two masses.
What does it actually mean when we multiply?
At the beginning there were the natural numbers. If you only consider natural numbers, you can view multiplication as a shorthand for addition:
$$3 \cdot 2 = 3 + 3$$
In general $a \cdot b$ means sum $a$ a total of $b$ times, or (interestingly) sum $b$ a total of $a$ times.
After this came rational numbers (or fractions) in the form $\frac ab$; and of course you define what does it mean to sum two fractions. But now how do we define multiplication? Clearly it's not a shorthand for addition now, since you can't add a fractional numbers of terms. So people said: define
$$\frac ab \cdot \frac cd = \frac{ac}{bd}$$
Why was it defined this way? It has something to do with preserving some properties we would like an operation called "multiplication" to have (that is, we want to preserve some properties that the "original" multiplication between natural numbers had)
And in the same spirit (albeit in a little more complicated form) we define what does it mean to multiply two real numbers (again preserving the same properties we expect multiplication to have)
The main thing to take from this is that multiplication is a purely mathematical concept: it is the extension to real numbers of a concept (really, a shorthand for addition) between natural numbers, and this extension is done in a arbitrary (albeit natural) way.
Now your question is: what does it mean to multiply two masses like $m_1 m_2$? And the answer must be: a priori there is no physical meaning. You're just performing an abstract mathematical operation on two real numbers. As it turns out, though, this "abstract mathematical operation" is useful to model the way the world works. One may say that the justification for its existence is the fact that works. Notice that this is not restricted to multiplication; in physics there are thousands of mathematical operations that are constantly applied, and the fact that they work (and can be used to model our world) is something which is not clear beforehand. There is a famous paper by Nobel laureate Wigner, beautifully named The Unreasonable Effectiveness of Mathematics in the Natural Sciences which also takes on this point, among other things.
So to recap: mathematics defines abstract concepts with no intrinsic (or physical) meaning. Physics then picks them up and uses them to build the world, and for some reason it works.