What about the explanation presented in the next paragraph?

> Consider how an exponent affects values between 0 and 1. Numbers close to experience a strong pull towards while larger numbers experience less pull. For example 0.1^2=0.01, a 90% reduction, while 0.9^2=0.81, only a reduction of 10%.

That's exactly the reason why it works, it's even nicely visualized below. If you've dealt with similar problems before you might know this in the back of your head. Eg you may have had a problem where you wanted to measure distance from 0 but wanted to remove the sign. You may have tried absolute value and squaring, and noticed that the latter has the additional effect described above.

It's a bit like a math undergrad wondering about a proof 'I understand the argument, but how on earth do you come up with this?'. The answer is to keep doing similar problems and at some point you've developed an arsenal of tricks.

In general for analytic functions like e^x or x^n the behaviour of the function on any open interval is enough to determine its behaviour elsewhere. By extension in mathematics examining values around the fundamental additive and multiplicative units \{ 0, 1 \} is fruitful in illustrating of the quintessential behaviour of the function.