The reason for this is that when we consume more of a good or a service, the next unit of that product comes to have less utility for us (in general). If I have not eaten, the first sandwich will have a great deal of utility for me because I am very hungry and it will really help to satisfy my hunger. But after I've eaten two sandwiches, I'm no longer very hungry. A third sandwich would not have much utility for me. This is how decreasing marginal utility works.
So now think about how much I would pay for these sandwiches. For the first one, I might pay a lot. I'm very hungry and I really want the sandwich. I'd pay a little less for the second sandwich because I'm less hungry. By the third sandwich, I wouldn't pay much at all because I don't really need it that much. Thus, as the quantity of sandwiches goes up, the price I will pay for them goes down. This yields a downward sloping demand curve.