This is because all firms in perfect competition are by definition selling an identical (homogeneous) product.
If all firms are selling an absolutely identical product, there is no possible reason why someone would pay a higher price to Firm A than they would pay to Firm B. Because of this, any firm that raised its price would lose all its market share.
The curve is also like this because firms in perfect competition make no economic profit. If they drop their price, they will go out of business.
So, they can't raise their prices and they can't lower their prices. That means their demand curves are horizontal.
Here is a video explaining the concept:
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