I like Justin's answer but I would point out that (fortunately) sellers don't even need to know about Supply, Demand, and Equilibria to react and adjust their prices as the theory suggests.
Here is one of the stories I tell my students in Intro to Micro and which does not require sellers (or buyers for that sake) to have any understanding or abstract notion of S&D theory (if someone wants to add pictures, feel free to edit):
- Suppose supply increases, which changes the equilibrium from $(P_1,Q_1)$ to $(P_2,Q_2)$ with $P_2 < P_1$ and $Q_2 > Q_1$.
- At the initial equilibrium price of $P_1$, there is now a surplus with $Q_S(P_1) > Q_D(P_1)$.
- This means there are "a lot" of people ($Q_S(P_1)$) who would happily supply a unit of the good at $P_1$ (because their WTS is higher than $P_1$), but only "a few" ($Q_D(P_1)$) who would like to buy (because their WTP is lower than $P_1$).
- In this situation, should we really expect the price to remain at $P_1$? I.e., should we expect suppliers to keep posting a price of $P_1$? If that was the case, many producers would not sell at all. That can mean either that their inventory would start growing (if they are in a market where they have to produce before they sell), or that some willing producers will be left idle (if they are in a more flexible "sell as you go" kind of market).
- Either way, producers who do not sell at a price of $P_1$ will easily notice it. They could remain idle or exit the market. But most likely, they will try to do something about it and stay in business before considering exiting the market.
- In a surplus, many of the producers who do not sell have a $WTS < P_1$. In other words: they have some margin for action in the form of price reduction. Rather than going out of business, they will try to convince the few $Q_D(P_1)$ buyers to buy from them instead of their competitors. There are many ways to do that (increase quality,...) but in the short run, the easiest way is probably to decrease your price somewhere between your WTS and $P_1$.
- This is how prices start spiraling down towards the new equilibrium, simply because producers obsreve that they are not selling, and they try to do something about it by lowering their price (while still making a producer surplus and selling above their WTS).
- Notice that all producers need to know are prices and their WTS in order to behave "appropriately" (they also need to observe whether or not they sell). In particular, it is not required for them to think in terms of (aggregate) S&D, understand what an equilibrium is,... In this sense, the price adjustment mechanism can be a purely decentralized one.
By the way, you can tell a similar story where consumers are active in pushing prices down. In a surplus, consumers have some "market power". They see producers line-up at their door begging them to buy from them instead of their competitors at the current surplus price of $P_1$. As a consumer, what will you do in that situation? Will you stay "idle", pick you producer at random, and pay $P_1$? Or would you rather tell one of the producers "I'll buy from you instead of your competitors, but only if you offer me a lower price than $P_1$"?.
"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 story is similar and somewhat symmetrical for increases in demand.
When demand increases, staying at the old equilibrium price $P_1$ leads to a shortage. This means if we stayed at $P_1$, not all consumers get to buy a unit of the good. These consumers can stay idle (unlikely) or try to convince producers to sell to them instead of other consumers (more likely). In the second case, the easiest way to succeed is for consumers with $WTP > P_1$ to tell producers "I'll pay you more than $P_1$ (but still less than my WTP) if you promise you will sell your product to me rather than another consumer", which initiates an upward price-trend.
For the story where producers are active, under the shortage at the price of $P_1$ producers see consumers line-up for a chance to buy a unit of their good. Producers may think that all the consumers lining-up have a WTP of exactlyu $P_1$ (unlikely) or that some of them have a WTP of more than $P_1$ (more likely). In the second case, producers know they can increase their price --- at least ever so slighlty --- while still selling their unit of the good. This clearly raises their producer surplus and we should expect them to so, which again initiates an upward price-trend.