Why is it preferable, when comparing payment options, to sum the discounted cash flows instead of simply adding the cash payments?

Because it's not worth the same thing today as it is in the future. If you could take $1000 right now, or you could take $1000 in 2 years (and let's assume you're not in any hurry for it), you would definitely want it now. $1000 in 2 years isn't going to be worth the same thing.

Here's an example with some real numbers. You have the option of taking $10,000 now, or taking $10,500 in 5 years. At first glance, does the $10,500 seem worth more? Well, what if you could take your money and invest it at 6% a year?

Normally this is done by figuring out the present value of all future values. That puts everything at a value today where you can compare them. So we would want the present value of the $10,500, at 6%, for 5 years. That is like saying: how much would you have to have today, invested at that amount for that time, in order to grow into the $10,500 5 years from now?

And that answer would be only $7846. You need $7846 today to make it grow into $10,500 five years from now. That means it's "worth" only $7846 right now.

So the $10,000 you could get today is definitely worth more than only having $7846 today.

"Discounting" the cash flow is meaning finding the present value.