There's a sense in which it's totally arbitrary, like the alphabet. It could be any order, but the important thing is that we settled on a particular order.

On the other hand, there's a sense in which the order we use was the only one that made sense, or maybe one of two that made sense. Let's forget about division and subtraction for a second. Multiplication is just repeated addition, and exponentiation (or "order" if you use BODMAS) is just repeated multiplication. So it really only makes sense to go power-multiplication-addition or addition-multiplication-power. Then, we get parentheses (or brackets for BODMAS). The whole point of brackets is to say "DO THIS BUT FIRST," so it has to be at the start.

Now, subtraction and division are just shorthands for adding negatives or multiplying by fractions, so that's why they go in pairs with addition and subtraction being equal and multiplication and division being equal.

So why exp-muli-add? Well, we often want to write something like 3y²+2y+6. If we used add-mult-exp, then we'd need to rewrite this as:

(3×(y²))+(2×y)+6

So we pick the format that makes the notation simpler and easier to read.