Learn Arabic (Modern Standard): سمع الأسد بأذنين كبيرتين.

This is because of the little prefix بـ (with) attached to the word for ears. It is a preposition, and prepositions force the noun after them to change its ending. For dual nouns, the regular ـان suffix changes to ـين. Since the adjective (كبيرتين) must perfectly match the noun it describes, it copies the ـين ending too.

Where does the letter ت come from in the middle of the adjective كبيرتين?

The ت shows that the word it describes is feminine. The singular word for ear is أذن. In Arabic, body parts that come in pairs (like ears, eyes, and hands) are almost always treated as feminine words. To make the feminine adjective كبيرة (big) dual, you open the final ة into a regular ت and add the dual suffix ـين.

Does the feminine word for ear (أذن) change the gender of the verb سمع (heard)?

No, because the ears are not doing the hearing! The lion (الأسد) is the subject doing the action. Since the word for lion is masculine, the verb stays in its default masculine form, سمع (he heard).

Why does the sentence start with the verb سمع instead of the noun الأسد?