Maybe I got it?

A prime number has only two multiples, 1 and itself.

A composite number has more than two multiples factors.

A composite number is guaranteed to have a multiple factor that exists somewhere between 1 and the square root of that number.

The square root is the tipping point on the multiple factor balance beam. I'll try to write out some examples.

Lets take 16's multiples factors:

1*16
2*8
4*4
8*2
16*1

Once you get past the square root the numbers "flip around" such that you're re-showing that 2 and 8 are multiples factors.

Let's try 100.

1*100
2*50
4*25
10*10
25*4
50*2
100*1

Again we see that IF another multiple factor past the square root exists, it will have been detected prior to the square root.

I hope that this is the reason the square root trick works or I'm going to have to take it on faith and that just irritates me because it breaks my "don't use it unless you understand it" rule.