how does perfect competition lead to demand curve of the firm being perfectly elastic?

In a market where there is perfect competition all firms involved are selling the same product (commodity), and because there is no product differentiation they must also charge the same price.

If you look at the graph of the demand curve for the individual firm, you'll see that it's perfectly elastic (horizontal) at the particular equilibrium price that the firm must charge.

The perfectly elastic demand curve also illustrates that no matter how much of the firm's product is demanded (theoretically) the price level will not change. This happens because in perfect competion there are so many firms involved that a change in demand for one firm will not be enough to change the demand and equilibrium price level for the entire industry.