Why does it have such significance?
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The answer involves calculus. Without calculus you can’t really appreciate the importance of e. Note the ancient Greeks knew about pi but not e.
The most basic important differential equation is dy/dx = cy for a constant c: something changes at a rate proportional to itself. The proportionality constant is c. Many ordinary differential equations that can be solved explicitly turn out to be related to equations like that.
If dy/dx = cy and y(0) = y*_0_* (the initial value), then the solution to that differential equation is y(x) = y*_0_*e^(cx). All of these are *exponential* curves (increasing if c > 0, decreasing if c < 0, and constant curves if c = 0).
In particular, to take the simplest case, if dy/dx = y and y(0) = 1 (so the proportionality rate is 1 and the initial value is 1) then y(x) = e^(x).
There are connections between e and trigonometric functions, but that involves further calculus plus complex numbers, so it is not as basic a reason as what I wrote above.
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