The way that economists explain this is by saying that an increase in demand means that more people are bidding for the good.
So you can think of this like an auction on eBay. When lots of people want some good, they bid it up and the price goes higher. Economists say that the prices of goods in stores works the same way.
In practice, what happens is that stores start to see that the products are selling really quickly at a given price. Then they try to raise the price to see what happens -- to see if they can get away with it. If they can, prices rise and the rise was caused by having more people buy the products.