As a mathematician who taught "Business Calculus" to college kids for a couple of semesters (yes, it was as awful as it sounds), here's the deal:

Continuously compounding interest is a useful mathematical abstraction that gets used all over the place. At its most basic, all you really need to understand is how to use the formula A=Pe^(rt). The variable r is the interest rate, t is the time, P is the amount of money you start with, and A is the amount of money you end with. Without getting into calculus and limits, that's really all there is to it.

But that's perhaps not a very satisfying answer, so let's get just a little more complicated. Let's start by looking at what happens if you compound your interest once. If you start with P dollars and your interest rate is r, then after calculating your interest and adding it back to your total, you get a final amount of A=(principal) + (principal)*(rate). That is, A=P+Pr, or A=P(1+r).

Now if you compound your interest yearly, the formula is A=P(1+r)^t, where t is the number of years. You're just going to have to trust me on this, because I'm hungover today and having an impossible time actually deriving this formula, I may come back and add an explanation at some point when my brain is actually working.

Finally, we can compound interest more often, maybe twice a year or quarterly or, more generally, n times a year. In this case, the formula is A=P(1+r/n)^(nt), and again, you're just going to have to trust me on this. It's not too hard to derive (and possibly your textbook even has the derivation somewhere), but I'm not coming up with it, because of the aforementioned hangover.

Now here's where the interesting bit of continuously compounded interest comes in, and why I mentioned limits several paragraphs ago. If you take that n (the number of compoundings per year) and you increase it, the (1+r/n)^(nt) approaches e^(rt). You can make this precise with limits, but you can also check this experimentally by choosing some r and some t, maybe r=0.05 (5% interest rate) and t=7 (7 years), and see what happens to the resulting number as you plug in larger and larger values of n. Maybe try n=10, n=100, n=1000, n=10000. You'll find that you get very close to e^(rt), which seems like magic, but it's really just limits.

So yeah, that's really all there is to it, the number e is weird and cool and continuously compounding interest is just what happens when you take your formula for interest compounded a few times a year and let the number of compoundings per year go to infinity!

You'll notice I never talked about any actual economics or anything, that's extremely not my field, money is fake, it's all stupid, death to capitalism, you know the drill.

Oh, one more fun fact, that same formula A=Pe^(rt) can also calculate inflation if you know the rate of inflation. It's the exact same idea and that's exactly how people translate how much money was worth in the past to how much it's worth now.