> it jumps from "use cos(t) + i sin(t)" to "use e^it" with zero explanation about why it's related to 2.718281828...
It really isn't, beyond the fact that exp(1) = 2.718…. Then, by definition exp(it) = exp(1)^(it) = e^it. It isn't much more than a notational artifact as far as I'm concerned. For that matter, raising any base b to a power is equivalently just scaling it by ln(b) and plugging it into exponential function, that is, b^it = cos(ln(b) * t) + i sin(ln(b) * t).