What does it mean that in a free market that buyers compete against other buyers for goods and services and that sellers compete against other sellers?

That is economics. Also known as 'Supply and Demand'.It means that the buyers are competing against each other like if one person wants the new Lamborghini and another also wants it, but there is only one in stock, they will have to compete not by fighting but by business, same with the sellers. If there is this new hat and a lot of sellers have it, the sellers will have to compete to sell the hats at the best price, but also earn a profit. Who ever has the best price will attract the buyers.

(P.S. A free market is a market in which the Government does not get involved between the buyer and the seller. That is called Capitalism. In some countries, the Government puts the price. So the Government is getting involved. That is called Communism:)