By what means do prices change? The classical supply and demand theory states that prices and quantities of a given good (in a perfectly competitive market) are determined by the supply and demand of that good, a shift in the demand curve, for example, would change the equilibrium prices, but how exactly does that work, can't imagine a firm owner saying 'Ok guys, the supply curve for our product shift to the right, time to raise prices"

I can grasp how a price above the equilibrium would go down (the producers would not sell, so they lower the price because less profit is better than no profit)

But how can an increase in demand drive prices up?
The simplest method would be to watch how quickly your inventory is depleted.  If your baseline is that you sell 1000 widgets a day at a dollar per widget and suddenly you're selling 1250 widgets a day (or selling out of the 1000 widgets your factory can produce a day by early afternoon), it is reasonable to believe that the demand curve has shifted to the right (there demand at a price of $1/ widget has risen by 25%) and to respond by raising prices.  From that same baseline, if you're suddenly only selling 750 widgets a day, it is reasonable to assume that the demand curve has shifted to the left and that you need to respond by lowering prices.  

In the real world, think of the hot toy every Christmas.  If your supply of Tickle Me Elmo dolls fly off the shelves minutes after you get them and grown adults are fighting each other in the store to get their hands on one, it's a pretty safe bet that you're selling them at below equilibrium prices.  If the Furbies on the same shelf sit there for months without selling, you've probably priced them above the equilibrium price.

Of course, in reality, stores may have additional methods.  They can look around at what other people are selling items for.  They can do market research to survey customers and potential customers to determine the demand.  Online, they can show people different prices for the same product and see how that impacts purchase decisions.