Why is exponential decay/growth so common? What is so significant about the number e?

Any system where the rate of growth is proportional to its current size can be represented by an exponential function. This is essentially the definition of an exponential function:

y = e^x <-> (d/dx)y = e^x = y

So, for example, the population rate of a species in an environment with sufficient food and no predators can be represented by an exponential function, because the rate at which new animals/cells is created increases linearly as the population (or number of potential parents) increases.

Additionally, if you have a bank account with interest, that also can be represented by an exponential function, since the rate at which you gain money from interest increases as you get more money.

So "e" pops up anytime there is continual exponential growth, i.e. when the rate of change of a system grows continuously with its size.

I think this guy's explanation describing the intuitive nature of "e" and exponential functions is really good if you're looking for more detail