The reason for this is that the firm in perfect competition is a price taker whose marginal revenue will always be equal to the price of the good or service.
In perfect competition, the price of a good or service is completely unrelated to the number or units of the product that are produced and sold. The firm can sell any quantity of the product that it wants at the same price. Because of this, the marginal revenue for each unit sold is always the same. Since the marginal revenue is always the same as the price, and since both numbers are always the same, the marginal revenue will also be equal to the average revenue for all units sold.
In perfect competition, there is only one market price no matter how many units are sold. For this reason, the MR will always equal the AR.