From what I gather, price fixing works by getting all the relevant actors in a certain economic sector to agree to fix the price of the goods they sell at a certain, higher-than-normal price, so that the consumer is forced to buy overpriced goods.
My question is: how does this even work in real life? If the goods are overpriced, what stops a new seller from offering his/her goods at the real (lower) price, rendering the price fixers unable to profit from their scheme?
In the real world, "price fixing" is very difficult to accomplish. In most industries, it would be very easy for a new seller to enter the market and offer their goods at the lower price, stealing the customers from the "price fixing" sellers. For these sectors of the economy, "price fixing" is not an option unless the "price fixing" sellers can convince/coerce any entering sellers to commit to the existing fixed price.
Ordinarily, "price fixing" occurs in industries with either a monopoly or an oligopoly.
In a monopoly, there is only one seller. So, they alone are able to dictate the price of the good or service.
In an oligopoly, there are only a few sellers. These few sellers can easily conspire with each other to elevate the price. However, even then, there is always the danger that one of the sellers will eventually drop their prices and try to steal the consumers from the other sellers.
So you can think of this issue in a game theoretic sense. Think of a simple case where each oligopolist has two strategys; collude, or price war. Though it may be Pareto optimal (for them) to collude, there is a prisoners' dilemma-esque incentive to deviate and undercut one's competitor. Thus, even without entry, collusion may be difficult to sustain.
Naturally, price fixing and collusion interests many economists and the resulting literature is huge.