I think it's helpful to think about sums for a bit, and then come back to products.
With sums… we’re all happy with the idea that you can add up a list of 2 things, or 3 things, or more (so e.g. the sum of the list $(x,y,z)$ is $x+y+z$). And after a moment’s thought, we’re happy with the idea that you can add up a list of 1 thing, eg $(x)$, and get just $x$ itself. And it takes another moment’s thought, but it's still pretty intuitive, to decide that you can add up the empty list $()$, and that its sum is $0$.
(“I went shopping today, with nothing on my shopping list. How much did it cost in total?”)
Now, play exactly the same game with products! It's just as easy until you get to the empty list. For some reason, it's much less intuitive at first (at least it was for me, and I think it is for most people) that one can make sense of the product of the empty list, and that it should be $1$. But every argument I know to justify why the sum of the empty list should be $0$ also works as an argument that its product should be $1$! (See the arguments about making exponents work nicely, etc, in the earlier comments and answers.) And, of course, once the lightbulb flashes, it suddenly becomes completely clear and natural that the product is 1, and that that's not just a convention, and how could anyone possibly think otherwise!? :-P
…unfortunately I don't know an example for products that’s quite as intuitively convincing as the shopping list example is for sums. The best I can do: at the checkout, they apply a bunch of taxes and/or discounts. Each one of those is a multiplier: e.g. 7% sales tax multiplies my total by 1.07, a 20% discount is a multiplier of 0.8, and so on. To get the overall multiplier from all the taxes/discounts currently in effect, you just multiply the list of them together (of course)! But if there are no taxes or discounts at the store today, what's the overall multiplier on my shopping?