Since a perfectly competitive firm can sell as much as it wishes at the market price, why can the firm not simply increase its profits by selling an extremely high quantity?

In perfectly competitive markets, producers do not have a say in the equilibrium price. Instead, this is determined by the market and with consumers who have what is considered "perfect information." If a firm is to sell its quantities at the market price at quantities higher than the equilibrium quantity, they will incur losses. With each unit produced, more costs are incurred by firms and producers. This is called the marginal cost of production. Just because a firm is selling at an equilibrium price, does not mean they may sell and produce extremely high quantity. Over time, this will lead to a rise in the average total cost, and again, the firm will begin to see losses.