The major way in which this happens is that the demand for the firm’s products becomes more elastic. In addition, it causes the demand for the firm’s products to drop.
In the short run, a firm in monopolistic competition can earn economic profit. When it does so, however, it attracts more firms to its industry. These other firms see that profits are high and they want a piece of those profits. There are very low barriers to entry so soon the other firms enter the market.
When the other firms do this, consumers have more choices. This will typically reduce the demand for the products made by any one firm. When demand decreases, we know that (all other things being equal) the equilibrium quantity and price will also decrease. If the prices that a firm charges drop and it sells a lower quantity, its profits will, of course, drop.
In addition, a firm’s demand will become more elastic. If a firm tries to raise prices to preserve its profits, consumers will simply switch to another firm. This makes it harder to make a profit as well.
For these reasons, the entry of new firms into the market drops economic profits to zero in the long run.