How do properly functioning markets maximize consumer and producer surplus while optimally allocating resources? Explain

A market maximizes consumer surplus and producer surplus by ensuring that quantity supplied equals quantity demanded. This means everyone who wants to buy/sell the good at that price is able to and there is no shortage or surplus. Thus prices are efficient.

If the market is competitive and healthy, then the outcome will maximize total surplus setting price equal to the intersection of supply and demand. When this occurs total surplus becomes the entire area between the two curves to the left of their intersection. A healthy market thus maximizes total surplus and that is how prices are determined which by default determines consumer and producer surplus.