Why would an increase in demand for a good cause an increase in price for a good?

Imagine that there are 10 apples to be sold at $1 each and that there are 10 people wanting to buy (demanding) an apple each. Now imagine there is an apple health-craze in this country and so now there are 30 people wanting to buy an apple each. Suddenly there are not enough apples being sold to meet the demand for apples. Always have in the back of your mind that "the free market always wants to be in equilibrium (which just means demand = supply)". So, in theory, the way the market fixes this disequilibrium (unequal demand and supply) is by raising the price. This naturally cuts off some of the people wanting to buy an apple since they cannot afford it anymore, and so the price will rise until 20 people are unable to afford an apple anymore and then there will once more be 10 apples supplied and 10 apples demanded. Realistically this makes sense as well. If two people are "fighting" over a good being sold, they may offer higher and higher prices to the seller until one of them can no longer compete and the highest bidder would win and buy the good at that new price. It is the same in a market, just usually with more people vying for goods. From the suppliers' side as well this makes sense: Suppliers are able to raise and raise their prices and still sell the same number of goods, increasing their returns. This is often what sellers want to do to increase profit.