This is the closest explanation I can find, although I still don't fully understand.
We know what’s happening: division is subtracting an angle and shrinking the magnitude. By multiplying top and bottom by the conjugate, we subtract by the angle of (1-i), which happens to make the denominator a real number (it’s no coincidence, since it’s the exact opposite angle). We scaled both the top and bottom by the same amount, so the effects cancel. The result is to turn division into a multiplication in the numerator.
What I understand so far is from a pseudo-example of real values. If you have an expression that is 1/5 = x, then if you multiply the left by c (in this case, 5) to get it to 1, you have to multiply the right by an equivalent value to get that side to 1. Extending the analogy to complex conjugates means that since (e^ix)(e^-ix)=1, you can just multiply each side by the complex conjugate to get both sides to one.
This implies that if two sides of a given expression are equal, then their complex conjugates are equal. Why is that true?
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$\begingroup$I'm going straight to the last question:
If two sides of a given expression are equal, then their complex conjugates are equal. Why is that true?
The idea here is that each side of the expression is a complex number. That is, there are real numbers $a,b$ such that each side is equal to $a+bi$. The conjugate of that number is $a - bi$. So, the conjugate of each side, is equal to the number $a - bi$.
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