Why does it have such significance?
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Suppose you have a 100 gram plant, and it grows by 100% in one month. At the end, you have a 200 gram plant.
But suppose it first grows 50% in half a month, and then 50% in half a month. If you compute it like this, it ends up bigger, because the first 50% of extra plant also grows in the second half: the growth compounds on itself. You ultimately end up with a 225 gram plant.
What if you split that 100% instead into “first 25%, then 25%, then 25%, and finally one last 25%”? Now the compounding growth gives you a slightly over 244.14 gram plant. What if you split into ten 10% one after another. A slightly over 259.37 gram plant. How about a hundred consecutive 1% increases? A slightly over 270.48 gram plant.
A million 0.0001% increases? Slightly over 271.828 gram plant. Is this number starting to look familiar? Yup, it’s getting closer and closer to e times our starting 100 grams.
This is one way to think about what e means. It’s a conversion coefficient between single-step growth, and continuous growth that builds upon itself: doubling turns into multiplying by e when split evenly across infinitely tiny steps that compound. Thus it is a very fundamental number when dealing with any sort of continuous compounding growth (or in other words, continuous growth that is proportional to the size of the original thing).
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